So earlier this week I read this post by Andrew Stadel and was challenged to approach the rules of exponents in a more perplexing way. First of all, starting with the mistakes and what kids already know about exponents was a great launching point for this lesson. I made my own 4 question mistakes not really wanting to get into exponents of zero, negative values or fractions just yet.
The following statements are all INCORRECT.1. Identify the mistake(s).2. Correct.3. Justify (show) your reasoning.We spend a large majority of the time on question 1 in the last hour and somehow managed to get to fractional exponents. Different from earlier classes today, I started my last class by saying there are two levels of answering these questions:
1. 43 · 42 = 165
2. (34)2 = 36
3. x3 · x2 = x6
4. x2y3 = (xy)5
1) You grab a calculator and find the value of each side and say "These numbers are different, so it's wrong".For number 1 I had students say that the value should be 45, 210 and 322
2) You apply what you already know about exponents and come up with an explanation that someone who knows less than you about exponents can understand.
The fun came in asking students to explain why 45 was the same as 210 in visual form. Most at first said that because you take half the base number you must double the exponent but we needed to know why to develop their understanding.
One student explained by pairing up the two's to make 4's by multiplication and stating there were 5 groups of these pairs--the whole class applauded his explanation after gasping that they got it.
So then we went to why 45 was the same as 322 and a girl student did this (again applause from the class followed):
Realizing we were getting dangerously close to logarithms in my 6th grade Algebra class I pushed the envelope and asked what the exponent would be in 8x = 45 and they jumped right in!! Some students naturally assumed it was 2.5 and stopped but others weren't satisfied.
A girl student in class offered to come up and draw a picture of what she thought was happening and we got this to look at as a class:
Her picture was an excellent launching point for discussion in thinking about what to do with left-over factors when we try to group to get factors of 8. Her mistake actually brought out a big difference in fractions vs fractions as exponents when another student tried what she said in his calculator. The class came to consensus that for the exponent, since it would take 3-2's to make another factor of eight the exponent should actually be 3 1/3 instead of 3 1/4.
Wow. Great class, fully engaged students and some great thinking going on driven by the students.