Thanks for putting this together Holly. So much of my math education never came close to this kind of framing and explanation. It was all zoomed in on one specific type of calculation with no context. Wish I had this when I was in school, but I'm saving all of it for my kids.
You're welcome! I'm glad you're enjoying it. These are hella fun for me to write and don't take a ton of energy or effort, and they are also huge mood-boosters. I was a little worried people would get sick of them, since a lot of people are either scared of math or just flat don't like math, but by the numbers they're actually doing ok. :-)
This isn't particularly essential to your explanation of TFToA, but I want to ask because it always comes up when I go over exponents with my students: How you would convince someone that a^0=1? I've got my standby explanation, but I'm curious how you would go about it.
Depending on context and the student, I have two usual explanations.
One is place value. In the integer 9876, 9 = 9 x 10^3, 8 = 8 x 10^2, 7 = 7 x 10^1, and 6 = 6 x 10^0.
The other is the multiplicative identity. How 1 gives us back the number, and an exponent of 0 is 0 copies of whatever number we're dealing with -- the product of no numbers at all. That one always sparks an interesting conversation. :)
I'll typically use your second approach, but I'll indicate that we sort of baked in an assumption about the starting point of the multiplication. The most common answer to when we have 0 copies of the base is that we just end up with 0, but I'll show them that we didn't actually "start" from zero when doing the exponential operation, because it's repeated multiplication, multiplying by zero would give zero. What we did was assume that our starting value was 1, so if we have zero copies of our base, then we're just left with our starting value. I tie it back to the point that when everything cancels out of numerator or denominator, you aren't left with 0, you're left with 1. (I also like this because it goes along with the initial condition constant that comes up in exponential equations).
Then, I'll reiterate the point when we get to how exponents cancel if we have exponents on the same base in the numerator and denominator. By that point they've already know that a/a = 1, and through a few examples, they've induced that exponents subtracted when identical bases are divided. This usually works in making it apparent why having an exponent of 0 reduces to 1.
I've never used the place value argument, though I should--it's a good one.
I like yours! With only two exceptions (bright homeschoolers who advanced beyond their parents), my tutoring has always been people who were struggling, and the place value argument has been anchoring and helpful for them. But I like yours a lot!
Thanks for putting this together Holly. So much of my math education never came close to this kind of framing and explanation. It was all zoomed in on one specific type of calculation with no context. Wish I had this when I was in school, but I'm saving all of it for my kids.
You're welcome! I'm glad you're enjoying it. These are hella fun for me to write and don't take a ton of energy or effort, and they are also huge mood-boosters. I was a little worried people would get sick of them, since a lot of people are either scared of math or just flat don't like math, but by the numbers they're actually doing ok. :-)
This isn't particularly essential to your explanation of TFToA, but I want to ask because it always comes up when I go over exponents with my students: How you would convince someone that a^0=1? I've got my standby explanation, but I'm curious how you would go about it.
Depending on context and the student, I have two usual explanations.
One is place value. In the integer 9876, 9 = 9 x 10^3, 8 = 8 x 10^2, 7 = 7 x 10^1, and 6 = 6 x 10^0.
The other is the multiplicative identity. How 1 gives us back the number, and an exponent of 0 is 0 copies of whatever number we're dealing with -- the product of no numbers at all. That one always sparks an interesting conversation. :)
How do you do it?
I'll typically use your second approach, but I'll indicate that we sort of baked in an assumption about the starting point of the multiplication. The most common answer to when we have 0 copies of the base is that we just end up with 0, but I'll show them that we didn't actually "start" from zero when doing the exponential operation, because it's repeated multiplication, multiplying by zero would give zero. What we did was assume that our starting value was 1, so if we have zero copies of our base, then we're just left with our starting value. I tie it back to the point that when everything cancels out of numerator or denominator, you aren't left with 0, you're left with 1. (I also like this because it goes along with the initial condition constant that comes up in exponential equations).
Then, I'll reiterate the point when we get to how exponents cancel if we have exponents on the same base in the numerator and denominator. By that point they've already know that a/a = 1, and through a few examples, they've induced that exponents subtracted when identical bases are divided. This usually works in making it apparent why having an exponent of 0 reduces to 1.
I've never used the place value argument, though I should--it's a good one.
I like yours! With only two exceptions (bright homeschoolers who advanced beyond their parents), my tutoring has always been people who were struggling, and the place value argument has been anchoring and helpful for them. But I like yours a lot!
*Riemann has liked this post.
Seriously, I wish more math teachers would leverage the trees.
Excellent lesson, Holly.