After watching an episode of "The Universe" yesterday, the wheels beagn turning inside our fourth-grade daughter's head as she struggled to comprehend the vastness of space. The conversation went something like this:
Abby: "Mom, how far is a light year?"
Dad: "It's about 9.46 x 1015m/sec. Just round to 9.5"
Abby: "Okay, what does 1015 mean? What is that little number above the ten?"
Mom: I leave and get a sheet of paper and a pencil and proceed to write down the following:
100 = 1
101 = 10
102 = 100
103 = 1000
104 = 10,000
"Abby, do you see a pattern?"
Abby: " Hmmm...Well, it looks like the numbers get bigger." (Abby continues thinking about things.)
Mom: "What do you notice about the number of zeros in relation to the "floating number thingy?"
Abby: "Oh, they match. The number of zeros match the little number. You're adding, right?"
Mom: " Good, let's call the "little, floating number" an exponent. Also, you're not adding. Look at 102 = 100 and tell me what you think it means."
Abby: "Oh, it's 10 x 2. So, the answer is 20."
Mom: "No, it can't be 20 because I already told you it was 100. Keep thinking!"
Abby: "Okay, it's 10 x 20. No, that's 200. I get it! It means you have a ten, two times."
Mom: "Good, now you're on to something. So, 102 = 100. What does 103 mean?"
Abby: "100 x 3. No, that's 300. Oh, 10 x 10 x 10. I get it."
Mom: "Now, think about this one. What does 32 mean?"
Abby: "It's 3 x 2. So, the answer is 6, right?"
Mom: "Abby, look up at what you discovered earlier. Keep thinking!"
Abby: "Oh yeah! It means you have the number three, two times. So, it is 3 x 3 or 9, right?"
Mom: "Yes! Great job! Now, let's learn a new word. The "3" is called the base and what was the "floating number called?"
Abby: "The exponent."
Mom: "Right! So, can you explain how this would work regardless of what number you are using?"
Abby: "Hmm... Well, the exponent tells you how many bases you have and then you multiply. So, 43 means 4 x 4 x 4 or 16 x 4 or 64. Right?"
Mom: "YES!!!!!!!" (Shouts of joy fill the room.)
Abby: "Dad, so let's go back to the light year problem. Just how far is it?"
Dad: "Well, it's 9.46 x 1015 m/sec. So, we have this:
1 light year = (3 x 108 m/sec)(365 days)(24 hr/1 day)(60 min/ 1 hr) (60 sec/ 1 min)
Abby: "What exactly am I supposed to do with this? You want me to multiply all of that out? How?"
Dad: He proceeds to explain the dimensional analysis and how things simplify to 9.4608 x 1015 m/sec.
Abby: "So, that would be 9.4608 x 1,000,000,000,000,000. Right? Wow! That's pretty a huge number!"
After this, I continued giving her a few problems to simplify. She was really excited and thought it was actually "fun." Wow! I proceeded to tell her it does get more complicated when you start talking about negative exponents. In true "mom fashion", I just had to show her an easy example. So, I wrote 3-1 = 1/3. Again, her curiosity was sparked and we did a few more. All in all, I feel really happy. An educational moment presented itself and we ceased the opportunity to explore. Carpe Diem! I seem to be saying this a great deal lately.
Anyway, I feel whether you are homeschooling or in a regular classroom. It is important to not veer away from a learning opportunity. Did I ever imagine I would be discussing exponents with our fourth grader? No, well, maybe. As teachers, we should be aware of teachable moments and capitalize on them. So what, if it deviates away from the "current lesson" or topic. Take a chance and go with it! Everyone may benefit in the end.