Today’s theme is polynomials: multiplying, adding, subtracting, and dividing them (simplifying rational expressions).

## Today’s Files

- Notes (and corresponding video)
- Homework

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Today’s theme is polynomials: multiplying, adding, subtracting, and dividing them (simplifying rational expressions).

- Notes (and corresponding video)
- Homework

We did some more work with Linear vs Exponential models today, and the homework is more of the same. We’ll start reviewing the unit as a whole tomorrow, with our planned Unit 9 test on **Friday, May 24th**

- Homework: HW 9.18 – More Linear vs Exponential Regression
- Lesson video – Identifying a table as linear or exponential

- S.9 – Exponential functions over unit intervals
- S.12 – Exponential growth and decay: word problems
- S.13 – Compound interest: word problems
- S.14 – Continuously compounded interest: word problems
- Review modules from Algebra I level

Today’s assignment is about deciding whether a linear or exponential model is most appropriate for a set of data. The short version is make a scatterplot and examine the picture. If it looks straight, use a linear model. If it looks curved, use an exponential one.

- Homework: HW 9.17 – Linear vs Exponential Regression
- Lesson video – Identifying a table as linear or exponential

More of the same today as yesterday, only this time working with *exponential* regression instead of linear.

- Homework: HW 9.16 – Exponential Regression
- Notes
- Lesson video – Another example of getting a regression equation from the calculator.

Not all linear or exponential relationships are as perfectly defined as we’ve seen. Sometimes the fit of a line or a curve is a bit more approximate, and finding the equation for these functions is a bit more tricky. Fortunately, your calculator provides you with a helpful tool in doing just this: **Regression**. The instructions for getting your calculator to do this are in today’s notes. They’re a little tricky (lots of button pushes), but pretty straightforward.

- Homework: HW 9.15 – Linear Regression
- Notes
- Lesson video – Instructions on entering data into the calculator (follows the notes sheet above!)

More with linear and exponential models today, specifically with *interpreting* what the parameters (slope, y-intercept, growth rate) mean in the context of the scenario they model.

The big takeaway is this: for both linear and exponential models, the *y*-intercept is the **starting point** or **initial value**. For a linear function, the slope is the **unit rate of change** – the change in whatever *y* measures per one unit increase of whatever *x* measures. For an exponential function, the value of *b* always represents the **growth or decay rate** – typically expressed as a percent. You’ll need to set *b* = 1 + *r* or *b* = 1 – *r*, then solve for *r* to get the percent change.

**Don’t forget to bring your Chromebook and calculator every day for the rest of this week!**

We’re in the last phase of Unit 9, which starts off by reviewing the difference between linear and exponential equations and deriving the formulas for these equations from a table of values.

- Homework: HW 9.13 – Linear vs Exponential Functions
- Notes
- Lesson video 1 – Identifying a table as linear or exponential
- Lesson video 2 – Getting a linear equation from a table
- Lesson video 3 – Getting an exponential equation from a table

I’ve decided to push the quiz a day to give us a little more time to review. Your homework is twofold:

- Complete these three IXL modules to a score of 80 or better.
- S.12 – Exponential growth and decay: word problems
- S.13 – Compound interest: word problems
- S.14 – Continuously compounded interest: word problems
- Note that S.13 and S.14 require solving for
*t*for some problems. For these, I advise using the Desmos calculator, enter**y = the compound interest equation**in one line and**y = the target dollar value**in another line, then looking for the point of intersection (and converting it into years/months)

- Complete HW 9.11.5 – Quiz 2 Review Supplement. We did this in class, but if you weren’t here you should work on this tonight! Answers can be found here.

**8th period: **Please watch this video to introduce the big idea behind today’s lesson (late in the video, the term “gradient” refers to the slope of the line tangent to the curve at a point), then watch this video from 14:30 to 18:15 for some examples of how to use this formula. Your homework will be the same as below.

We expanded on compound interest today, extending the concept to the idea of “compounding continuously,” which in turn led us to the fascinating mathematical constant of *e*. If you’re interested in learning more about *e*, click the link!

- Homework: HW 9.10 – Compounding Continuously
- Notes
- Lesson video 1 – The is a Numberphile video explaining what
*e*is and some of the interesting things it does. - Lesson video 2 – Skip to the 14:30 mark for information on compounding continuously, but the first bit is a great review of other compound interest problems.

A special case of exponential functions today: Compound Interest

- Homework: HW 9.9 – Compound Interest
- Notes
- Lesson video 1 – using the compound interest formula (the video’s notation is a little different from ours, but the method is the same)
- Lesson video 2 – finding the starting value for a compound interest problem

Continuing with the unit on exponential functions, today we spent some time working on creating functions that model real-world data.

- Homework: HW 9.8 – Exponential Function Modeling
- Notes
- Lesson video 1 – a review of percent change problems from Algebra 1
- Lesson video 2 – setting up an exponential function equation to solve for the base (starring Mr. Kirk’s hand!). This video works through the second example from the notes linked above.

**S.9 – Exponential functions over unit intervals****S.12 –****Exponential growth and decay: word problems**- Review modules from Algebra I level

We finished our lesson on understanding marginal tax rates, and your homework is to work through a few more examples. You can use the online tax calculator found here to check your work.

- Homework: HW 9.7 – Understanding Marginal Tax Rates
- Notes
- Lesson video: Calculating marginal tax rates

- Review modules from Algebra I level

Your first quiz of Unit 9 is tomorrow. Remember that this is a **no calculator** quiz.

- Homework: HW 9.5 – Quiz Review 2
- Quiz Review Answers
- Notes
- Lesson Video Archive
- Simplifying Exponential Expressions
- Exponential growth/decay functions

- Review modules from Algebra I level

Today, we revisited the exponential growth and decay functions we first saw with the M&Ms activity on Wednesday, and we formalized our notation and vocabulary a bit.

If you’re here to watch a lesson video **in class**, you can find it here.

- Homework: HW 9.3 – Exponential Growth and Decay
- Notes
- Lesson video – This video refers to a specialized version of the exponential function formula we discussed on the notes, where there is a specific % increase or decrease, but the overall content is the same. You can also stop watching at around the 7-minute mark, as it goes on to examples we will not do.

- Review modules from Algebra I level

Today, we reviewed the various shortcuts for simplifying expressions with exponents, and tonight is some practice of them.

- Homework: HW 9.2 – Review of Exponent Shortcuts
- Review of exponent shortcuts
- Lesson video 1: Multiplication Rules
- Lesson video 2: Division Rules
- Lesson video 3: More Division and Negative Exponents
- Lesson video 4: Rational Exponents

- Review modules from Algebra I level

Today, we reaffirmed that the Expected Value is a **long-term** average. Meaning the results of a small handful of trials are unlikely to produce the same average as the expected value, however increasing the number of trials will produce averages more in line with what you expect.

Don’t forget that our test on this unit will be on **Thursday, April 11th!**

- Homework: HW 8.12 – Understanding Expected Value
- Lesson video: Calculating Expected Value (this is the same video as what was posted yesterday)

We did some more practice with calculating Expected Value today, and the homework is to continue that work.

Our test on this unit will be on **Thursday, April 11th**. If you will not be in school that day, **you must make up the test before you leave**. NO EXCEPTIONS

- Homework: HW 8.11 – Expected Value Practice
- Lesson video: Finding expected value from a probability distribution

Permutations and combinations tend to be tricky to figure out when you first encounter them, so we spent some more time looking at examples, including a modification to the concept to find arrangements of letters in a word.

- Homework: HW 8.6 – More Permutations and Combinations Practice
- Notes: Permutations, combinations, and letter arrangement puzzles
- Lesson video 1: How to tell the difference between permutations and combinations (with examples)
- Lesson video 2: Arrangements of letters in a word

The next part of the “counting” part of this unit is understanding the difference between two common patterns of counting: permutations and combinations. In brief, permutations are used to count the number of **arrangements** of objects, specifically when order matters like with ranked lists, unique assignments, or sequences. Combinations are used to count the number of **groups** of objects, specifically when order *doesn’t* matter like with committees or batches.

- Homework: HW 8.5 – Permutations and Combinations
- Notes: More details on the difference between these methods
- Lesson video 1: The difference between these methods (including also factorial)
- Lesson video 2: Calculating permutations and combinations (note that you can just use your calculator, instead of doing this by hand.

We did some more work with understanding the nature of mathematical independence today, including making some graphs to help illustrate whether or not independence exists in a group of data.

- HW 8.4 – Illustrating Independence
- Notes: Illustrating Independence with Segmented Bar Charts
- Lesson video: Creating a segmented bar chart