Lesson 3.1.2 starts with a Silent Board game. This is where you can see the students who can think quicker than others, and figure out the rule by telling the missing output values when given just inputs and a few correct inputs. Take a look at the Silent Board game A for yourself, and students practice it if you want more practice. Students don't tell me the rule until all outputs are filled in. This was also practice from the previous nights homework.

Then students move to analyzing a table to figure out how the tree is growing. They are also given its height in years 3, 4, and 5, and are asked to explain how they get the height in years 2 and 7. Then they are asked to figure out the height of the tree the year it was planted. A lot of students assumed the answer was 9 feet, because that's the height for 1 year. I asked them, "is that the height when it was planted or after 1 year of growth?" That made them rethink their answer. Some students assume it's a proportional relationship especially when they predict how tall it it is in 50 years. Some estimated 250 feet because in 5 years it's 25 feet, so they multiply 5 by 10 and 25 by 10. Some estimated 205 feet because it grows 4 feet each year, and 50 times 4 is 200 and add the 5 feet for when it was planted. They test these predictions at the end of the lesson.

After the estimation, they completed the rest of the table. They are asked if it makes sense to connect the points. It ends up being a continuous graph, so it makes sense to connect it because it's growing in between each year at a constant rate. Students did mention it grew at a constant rate. Thankfully, they did say it was not a proportional relationship because it did not pass through the origin. I asked students to improve their answer if they said it didn't start at 0. James made a great comment, that yeah, it does start at 0, it's 5 feet at 0. I took that opportunity to mention he was displaying the common core standard of mathematical practice of using precision of language.

We then had some closure talking about the rule of the graph, and seeing if their estimation was correct or not and why.

In accelerated, I finally got to deploy a lesson I've been dying to try, which is Exponent Mistakes by Andrew Stadel (maker of our warmups, Estimation 180). It has 8 problems on it and they are all incorrect. They are instructed to explain the mistake, correct it, and justify why your new answer is correct.

Students work in groups almost daily in this class, so working for 20 minutes independently was a nice change. I then asked them to discuss what they came up with their groups after that. I then saw some lively conversations, some erasing, some changing of answers, and finally I prepared for volunteers to go up to the board.

So, I asked any student to come up and analyze any of the questions on the board. Alex showed why 2^5 does not equal 10 with a great explanation. Zoe attacked the 7^-2=49 mistake and said that 7^2 is 49, so 7^-2 can't be. She said 7^-2 is like 1/7 * 1/7 so it's 1/49. I then asked Harrison to discuss the 37^0 problem, and relate it to why negative exponents are fractions. He said as you increase the exponent you multiply by 7 again. As you decrease the exponent you divide by 7. This explains why 7^1 is 7 because 49/7 is 7. Then it shows why 7^0 is 1 because 7/7 is 1. Therefore, 7^-1 is 1/7 because 1 divided by 7 is 1/7.

One student attacked the problem 100^1/2 and said it was 1000. Another student at his group came up and wrote 100^1/2 is 10 because anything to the 1/2 is square root.

I wanted students to move away from the answer is this because of a certain rule. I wanted them to prove it through patterns and reasoning. I basically explained that what they were doing was algebraic proofs in a way. Prove it and convince the class why you are right, in a way that more than a few people could understand.

I asked the class, are you convinced? Can anyone prove that? So, as Mr. Stadel suggested, I asked students to calculate 100^1, 100^2, and 100^3. Alex D told us the answers. I then asked, how could this help us understand 100^1/2? Markos said that since each time you increase the exponent, there's 2 more zeros, then when you decrease the exponent you take away 2 zeros. But between 1 and 0 that's 1/2, so you take away 1/2 the amount of zeros, or 1 zero. There were a lot of a ha's. Then I asked for factors of 100. I said how else can we rewrite 10*10? They said 10^2. So I substituted that and it showed (10^2)^1/2. A lot of a ha moments and hands raised. Davin explained that you multiply the exponents so 1/2 of 2 is 1, and 10^1, is 10.

Nicholas had time to prove x^3/x^8 is 1/x^5 because he broke up x^8 as x^5 * x^3. Then canceled out the x^3 in numerator and denominator. I asked students what other way could we write that? They said x to the -3. I asked what rule does this prove? Gabriel stated that when you divide exponents with the same base you subtract the exponents. Great discussion!

We got through all the problems except for 1. This was a lot of fun and students genuinely enjoy coming to the board.

Tomorrow students have a substitute, so behave your best!

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