6.1 Recursive Routines
6.2 Exponential Equations
6.3 Multiplication and Exponents
6.4 Scientific Notation for Large Numbers
6.5 Looking Back with Exponents
6.6 Zero and Negative Exponents
6.7 Fitting Exponential Models to Data
I did the investigation as is from the book to explain the rules for zero and negative exponents. I then made two tables like that on page 367 to show the pattern of values from positive exponents down to negative ones.
I then did examples A, B, and C from the book with them. The questions themselves are on the PowerPoint presentation, but I was writing the solutions to them on the board. I spent more time on example B than any other – I showed them how they had to “go backward” without negative exponents first, and then showed them how easy it was using negative exponents for time.
For homework, I gave section 1 – 7 and 9.
Quiz Day
I will go over the homework from section 6.5 first, then go over the corrected review worksheets from the previous day. Then they will be able to take the following quiz which should take no more than twenty minutes.
Really a review day after I’ve moved rather quickly through this chapter. After going over the homework, I’m giving them the More Practice Your Skills worksheets for sections 6.3, 6.4, and 6.5 and having them do parts of them to turn in, which will be graded as a group.
6.3, 6.4, 6.5 More Practice Your Skills Review Answer Sheet
The next day I’m going to give a quiz on those three sections at the end of the period after going over the answers to the problems.
For the lesson, I pretty much followed the investigation and examples in the book, but I did them all on the board.
Example B is a bit of a leap for them. They know that the “A” means the starting value, but the kids have a difficult time thinking of the current value as the “y” part of the equation. Thus, writing 10000 = A(1 + 0.10)^20 is a real stretch for them. Then to know that they’ll have to divide both sides by (1 + 0.10)^20 becomes really difficult.
I tried to get over this by doing a problem forward first and explain that we multiply by (1 + 0.10) in order to find out what happens to the value in the future, so we’ll divide to find out what happens in the past.
Kids don’t see that the examples 5^9 / 5^6 = 5^3 make Cal’s answer in question #3 incorrect. Once the kids start dividing the coefficients of 10x³/2x, they then begin dividing the bases of the powers as well when they are numbers such as when writing numbers in scientific notation.
I’m going to put the book on the screen to do the investigation so that I can get the kids to understand what numbers look like in scientific notation – basically to define what a number in scientific notation should look like. I’d like to bring up “counterexamples” to help them get a good definition.
I’m then going to show the students how to set their calculators in scientific notation mode for them to use. However, many students don’t have them, so I’m not going to spend more than five minutes on that.
After that, I’m going to write lots of numbers in standard form and convert them to scientific notation by writing the products out so that they will see place value a little better:
250 = 2.5 × 100 = 2.5 × 10²
Then I’m going to lead them into multiplication of the numbers in scientific notation. I’ll write out the products as the book does in example A on page 357, but I’ll also mention that changing 125 to 1.25 means making one part of the result two decimal places smaller, so that they’ll have to compensate for that by making the exponent two decimal places larger.
The exercises incorporate enough of last sections work into it to keep me from having to spend a second day on 6.3. Also, since I am trying to get through the chapter and give a quarterly assessment with only nine days left in the quarter, I feel like I need to save as much time as possible. I’d rather spend more time reviewing all the concepts at once rather than spending extra days in each section. I have no idea if this is a good plan or not, but that’s my idea at the moment.
Homework # 1 – 5, 8, 10, 14 a,b
Decided to do the investigation with a PowerPoint presentation to make things easier.
I began with writing 4 + 4 + 4 + 4 + 4 + 4 + 4 and asking students how mathematicians would write this in a simpler way.
Next I put up 4 · 4 · 4 · 4 · 4 · 4 · 4 and asked them the same thing so that I could talk about how mathematicians like to write expressions in simpler ways.
Then I wrote some examples of the Multiplication Property of Exponents on the board and then had them derive the formula – for each example I wrote the problem out in expanded form.
I did the same thing with some examples of the Power Property of Exponents on the board.
I told the students that they would most certainly not remember the rules and would get them mixed up if they did so I stressed that when they encounter a problem where they weren’t sure what to do to write the problem out in expanded form.
Once those two were established, I went over more difficult problems with them such as distributive property problems for a few minutes before I gave them their work: 1 – 10, 13, 15.
Note: I heard a lot of 2x³ meant that both the “2″ and the “x” were raised to the third power.
My plan is to go over the homework assignment, then get through as much as the PowerPoint review as I can before giving the quiz on section 6.2.
I began the day going over each of the homework problems from the night before.
Question 1 I spent a great deal of time on using the overhead calculator while doing the problems on the board. I explained that we were trying to find what number multiplied by 16 gives us 20. To bring it back to the last chapter, I wrote down 16 * n = 20 on the board and asked them how to solve it. I did the same thing for the other “pairs” in the sequence. Then I did each on the calculator to show that they were all 1.25. I then wrote the sequence as a recursive rule with space next to it for the explicit rule which I told them that they would be able to do by the end of the day.
Question 2 I just did it on the calculator and then asked them what the textbook writers wanted them to figure out by looking at the terms (alternating between positive and negative).
Question 3 The most important one for understanding section 6.2 and percentage growth in general so I told them that if they don’t understand it yet, to make sure they try their absolute best to get it now. I spent some time (again) talking about the meaning of “per cent” so that they can change percents to their decimal equivalents.
Question 4 I called the initial $20,000 the 0th term, which apparently the book doesn’t use, so the answer in the book (for the fifth term) is the value after four years.
Then I did the following PowerPoint presentation with the group to get them ready for the explicit geometric rule and gave them problems 1-5, 9, and 12.
Objectives:
I’m going to begin with some examples of recursive routines of arithmetic sequences so that the students will have some familiarity with how to write rules recursively. I’ll remind them of “starting value” and “rate of change” as well as “term”. I’ll also take a recursive routine, write it explicitly in intercept form, and graph the line for them. Then I’ll tell them that all graphs are not lines, and that we’ll begin with another type today that by the end of the chapter they should be able to do the same routine with.
I’m going to have the students try to do the investigation by themselves and then I’ll go over the two examples in the book with them. The investigation and examples are in the PowerPoint demonstration, the investigation worksheet for the students to do is posted as well (it is just a table).
6.1 – Investigation and Examples PowerPoint
6.1 – Investigation and Example A Worksheet
Even more stuff. I’m not even sure if I’ll get to them, but I’m on a roll.
End of Class Quiz – (informal assessment)
6.1 – Quizzes (four versions based on the end of class assessment above)
