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A Composition function is just one function inserted as the “x” of another function.
Typically the notation for a composition function will look like this:
But you can also write a composition function like this:
and it means the same thing. Everywhere you see an “x” in the f(x) equation, instead of inserting a single number for that x, you are now going to insert the entire g(x) equation.
To Watch a Video of an example composition function worked out, click here.
If you are one of the many home schooling families using the AskDrCallahan math curriculum, you already know that we teach with college and life preparation in mind. So it might come as a surprise that we typically recommend parents give students open book and open notes tests.
When students sit down to take a math test, they either know the math, or they don’t. Having access to the textbook is not going to change their comprehension of the material. They THINK it will, but often that just is not the case.
We recommend open book/open notes because part of knowing how to do math is being able to figure it out using the information given. It has been our experience that open book tests are often more challenging than closed book tests because it ultimately comes down to “you either know how to do it or you don’t.”
Compare to Homework
When we make up our tests, we choose problems from the chapter review section that demonstrate the key concepts from the chapter and we assign them as tests. This approach is similar to picking homework problems for students to work. And if you think about what happens when students work their homework problems, they almost never get 100% of their homework correct and they have the book and their notes then as well.
Using the open book format strengthens students reasoning skills and teaches them to think about what they are doing, why the are doing it, and to dig in and give it a try even if they don’t know all of the answers. They will need to memorize some things, but rote memorization and endless math problems is not the way to excel in this subject. So give open book/open notes a try. Your students will like it, and they will learn more too!
Many homeschooling parents contact us when their math students are beginning Algebra 1. They are wondering what the difference is between Jacobs’ Teachers Guide and the AskDrCallahan Teachers Guide. In addition, they often wonder if they need a Teachers Guide and Solutions Manual for the Algebra 1 course. The answer is YES. and let me explain why:
For the AskDrCallahan Algebra 1 course, we recommend parents get the AskDrCallahan Teachers Guide and The Solutions Manual (Solutions manual ISBN 9780615315010). The Solutions Manual is a new (within the last year–2009/2010) product that Harold Jacobs (along with AskDrCallahan) put together.
It contains step by step solutions to every single problem in the book and functions as a significant improvement over his previous teachers guide (Teacher’s Guide 0716710757). The previous book was designed to accompany his textbook, but it only contained some solutions and they weren’t detailed so once this new book came out, we stopped carrying the Harold Jacobs teacher guide that was originally designed to come with the textbook.
Probably there are still articles in the blog archives that mention the old manual, but our Algebra website (http://shop.askdrcallahan.com/categories/Math/Algebra/Algebra-Full-Course-Bundle/) as well as our current article material should point you to the new manual. If you should have questions about which manual to purchase (or any questions for that matter!) please contact us, we will be glad to help. support@askdrcallahan.com
For the Algebra Course, we recommend our Teachers Guide (which contains the syllabus, tests, test grading guide, and additional resource material like activity suggestions and pacing information–download here). In addition to our teachers guide we also recommend the Solutions Manual( Solutions manual ISBN 9780615315010).
That said, many parents find that they can purchase just a textbook and our DVD set and do fine. The reason is because the dvd set contains a cdrom with pdf versions of our Teachers Guide and The AskDrCallahan “Solutions to Selected Problems”. This “Solutions to Selected Problems” is where AskDrCallahan has compiled detailed step by step solutions to only the problems we assign for homework. There are 60+ problems per lesson in this textbook and we do not assign all of them for homework, so we created this smaller manual for parents that intend to use our syllabus and therefore might not need the larger book.
To view more information about Algebra click here.
You can also download the Teachers Guide and other Algebra resources for free, by clicking here.
This month we are highlighting Algebra II with Trig Problem 57 in Lesson 2. 1 as our Problem of the Month for May 2010.
Recently, we received a question about problem 57, and the student was concerned because while they were getting the right answer, or so it seemed, their math problem was way different than the book. Confused, they turned to Homework Help, and here is what we told them. Hopefully it will help you as well.
The main difference between what you’ve done and the book’s answer is that you are trying to express your formula in meters, where the book is using kilometers. (which counts as your error only because the book specified that it wanted you to use meters).
So the short answers are : YES the book’s answer is correct and NO it doesn’t matter whether or not you write “C” throughout your problem as long as you know that you are dealing with Celsius.
Now let me see if I can explain:
Since the temperature increases 2.5 degrees Celsius (and you can just refer to it as degrees and not worry about Celsius vs Fahrenheit on this one since you stay consistently Celsius throughout the problem) for each additional 100 meters of depth, the temperature increases 25 degrees for every km. (1km = 1000meters; so that’s 10 “100 meters”, or 10(2.5) = 25 total degrees increase over the course of a whole kilometer. Another way to read it would be “you have to increase 2.5 degrees 10 times in a kilometer because you will travel a distance of 100meters 10 times. It is as if each 100 meters was one giant step, you are taking 10 steps). Does that make sense?
Therefore, if x = depth in km (which the problem specifically states to use kilometers and not meters for x), then x – 3 is the expression for the depth beyond 3 km . –and because we are using km instead of meters, our expression automatically allows for the “every 100 meters” so it is no longer necessary to divide by 100 meters.
Next, remember that the words “for every” means to multiply, so if we are increasing 25 degrees for every 1 kilometer after 3km, then we are increasing 25 degrees for every x – 3 of depth, and we write that 25(x – 3)
Since the starting temperature at 3km is 30 degrees C, then we have to start the equation with what we have. If we are sitting at 3km depth, the temperature is 30 degrees C. So we start there…
For more great Homework Problems worked out in this way, visit our Homework Help website. You’ll find links by course to all kinds of homework help and resources.
The Solutions Manual gives answers to all of the Homework Problems (and occasionally the example problems) provided in the textbook. Courses vary on the details of these answers, but typically there are only answers to problems contained in the solutions manual.
A teachers guide, however, provides aid to the teacher (parent) administering the course. The Teacher’s Guide contains tests, test grading guides, sample worksheets, pacing information and a general course overview. The Teacher’s Guide does NOT contain answers to homework problems
When you are deciding whether or not to use a math course, the Table of Contents is the best place to start (link to table of contents article). But once you’ve evaluated the Table of Contents, there are two other areas to consider:
Example and Homework problems in a textbook can give you a good idea of how the key concepts will be covered. Start by noticing the amount of example problems.
Then ask questions about the Homework Problems
These are just some of the questions you will need to ask about a course. You can also use these questions to evaluate what is important to you as a parent or student. Maybe you don’t need a bunch of example problems and you like how Book A covers the material. You find yourself enjoying to read it or you can easily understand their explanations. This level of connectivity with the text might override your desire to have tons of examples, so you’re willing to compromise. Then again, maybe you’d rather sacrifice connectivity with the text in order to have many examples. Still some others will wait it out until they find the course that mixes plenty of examples with connectivity.
The second thing to consider is the Tests. Tests give you a good idea of what the course creators feel are the important ideas to take from the lessons. It also gives you an idea of the difficulty level of the course. Some questions to ask about test:
What matters to you as a parent will vary with each student, with your own experience, and with each person. What course you ultimately get depends on your priorities as a parent, but ask the right questions about Example and Homework Problems and Tests.
Homework Help WILL work any problem you require step by step, but we are NOT intended to replace a good solutions manual. In fact, we are not ALLOWED to copy the book verbatim, since that would violate Jacobs’ copyright. What we do with step-by-step is work out problems in MORE detail than what the solutions manual provides for students that need it broken down further.
Homework Help is intended to clarify problems you come across when studying your subject. Whether it is math, English, history, or science, if you need help with that subject—we are there to be a virtual tutor and assist you on your way.
We also help parents and students find real world connections to their studies. We assist with project suggestions, project grading guidelines, ideas for how to encourage students to learn when you’ve hit a wall. We provide students with a place to ask all those “why” questions and parents a place to find all the answers to those “why” questions.
If you have trouble with your math problems, if you don’t understand why the solutions manual arrived at their answer, or if you need more detail than what the solutions manual provides, Homework Help is here to help you with all of these things.
We are not intended to replace good resources that are already available. We are simply another great resource you can use. But if all you need is someone to rattle off the answers to your homework problems, then by all means get the book that supplies answers. Get the solutions manual.
We do not currently have a system in place for doing the test grading for parents. If you are in need of this service, you may email us about that.
We will be glad to help you as we can.
However, we do provide parents with a detailed outline of how to test students. In our Teacher’s Guides for every course we sell, we include test grading sheets and sample test grading sheets to demonstrate our grading process. For most parents the Grading Guides we provide in our Teacher’s Guide are quite sufficient. You may download the teachers guide here. If you have questions about grading, test taking, or if you’re unsure if your process is correct, please send us an email. We will be glad to walk you through the process and get you pointed in the right direction.
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