Math chapter 3 non-positive integer exponents with materials
- 3. 2 4 21 22 8 23 20 Let’s take this block to represent 20 From the previous slide, let’s rename/label the next 3 bigger blocks…
- 5. 30 31 32 1 1 8 3 9 27 33 81 34 243 35 Let’s do the previous process to the powers of 3. The value of the blocks are…
- 6. 3−2 3−1 30 1 9 1 8 1 3 1 3 31 9 32 27 33 Let’s take this block to represent 30 and continue naming the succeeding blocks... The value of the blocks are…
- 8. The Rule: When we go from left to right, we are multiplying. But when we go from right to left, we are dividing. Since 20 is my unit, 2-1 says we divide by 2 one time. Likewise, 2-2 says we divide our unity by 2 twice. So a negative in front of our exponent tells us how many times we must divide.
- 9. Dividing by 2 is the same as multiplying ½. So 2-3 says to divide by 2 three times, or we can think of it as multiplying by ½ three times. This relationship will always be true. We see that 23 = 8, but 2 -3 = 1/8. What is the relationship here?
- 10. We can say then that negative exponents are how many times a number is divided. We can also think of this as multiplying by the reciprocal. So we have that if x is the base and –n is the exponent, then We also see that when we divide our base, x, by itself, we always get 1. So x0 = 1.
Editor's Notes
- #4: So we have these labelled to the right of 2. We want to know what these would be to the left. Let’s start from the largest and work our way backwards. What is happening to the exponents as we move to the left? They are decreasing by 1.”
- #5: Let’s keep going then. This next one must be 20. And we now this to be true because if this block is 2, then this is out unit, which has no power. Let’s keep going. What would this power be? Yes, it would be 2-1. What this mean, a negative power? We will find out shortly. For now, let’s continue to label.”
- #8: We do not have materials for the fourth powers, or to do larger powers of 2 and 3, but we could do the same thing on these charts.
- #10: Yes, they are reciprocals of one another. Is this relationship always true for any pair of positive and negative exponents?