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Algebra II with Trig Question – Is my Son Ready?

Question from a mom!

My homeschooled son struggles in high school math. He is very bright, and has taken Algebra 1 Algebra 2, and Geometry with Teaching Textbooks. However, he is not a “math mind,” and scored low in his yearly tests. He is going into his senior year, and I fear for his SAT/ ACT scores. He is very, very intelligent, but is not a good test-taker, and does not take naturally to mathematical concepts as they have been presented. How does your curriculum compare to Teaching Textbooks, and does your Algebra II/ with Trig. require that students be math geniuses already? Do you think that he could be introduced to your curriculum, and do well? 
I am a concerned mama…I would so appreciate a detailed/ helpful review/ analysis. Thank you so much.

Answer from Dr. Callahan:

We are college based where Teaching Textbooks are high school. What does that mean? Generally, taking our stuff is much better prep for ACT/SAT and college in general. Does ours require genius? Not hardly. In fact, to some it is easier because we teach concepts instead of formulas. 

For instance, if you put me in a  kitchen to cook, I can do OK with a recipe. But, I do not understand cooking and food concepts well enough to just throw something together (edible). However, someone who can cook and understands stuff knows the CONCEPTS and can do well. 

The ACT/SAT tests CONCEPTS. For a student who is smart, they thrive on concepts but hate being put in a box of rules (recipes) which are often part of high school math. So he might be better than you think he is!

Here is what I would do in your situation. 

 

  1. Get ACT/SAT coaching ASAP. This will help you and he determine where he stands. I would HIGHLY recommend someone who can work with him directly like Lauren (See http://www.askdrcallahan.com/sat-act-test-prep/) Note the discount code!!!! He is a senior – I would do this NOW. 
  2. As for math, I would have him evaluate himself. Go to the textbook website and have him take the quiz from chapters 1 to 8. He should do OK on the chapter 1- 5 tests. But if he struggles before chapter 5, I would have him take Algebra II with Trig. This is a fundamental course in the ACT/SAT. 
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Algebra II with Trig Ch8.3 #26

Question from Elisha:

 I just took the Trig. Final Test for your Alg. 2/ Trig. Course.  My mom and I were checking it and on problem 26 from section 8.3, my answer is different from the one you give in the answers.  I see that you add 45 degrees to the 25 degree angle and I didn’t.  I’m not sure exactly why that is and what I did wrong.  I have attached a photocopy of my work.  Could you tell me if my answer is wrong and if so why? Thanks for your help.  I have really enjoyed the course. J Elisha

 

Answer from Dr. Callahan: 

Elisha,
You are missing on  key thing…the angle between 15mph and 3.9 mph is not 25 degrees. 

The angle between due north and the direction is 25degrees. The angle between the direction of the boat and the direction of the water should be 70degrees. See example 1 again. 

Here is the solution worked out in a drawing. A2T Ch8.3#26

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Algebra II with Trig Ch3.4. Absolute Value

Question from Rachel:

Question involving absolute value in Algebra 2: 3.4.

Example 5 on page 233 states that “If X<0, then |x| = -x.

That really confused me. I thought that absolute value was just that – absolute! There are not supposed to be any negatives involved, right?

But obviously, there are, and I’m not sure why. I’d appreciate some help. Thank you!

Answer from Dr. Callahan:

I know it looks confusing – but note that they are NOT saying |x| = -x — but instead they are looking for a trick to help break up the complex function into pieces.

See top of page 232.

This is saying that anytime you have a function with absolute value – you know it is likely to be a piecewise function – in other words it will break somewhere like this one does

Homeschool Math Algebra II with Trig To get NORMAL functions, we need to break this into two parts (sometimes more) so we can get an equation like on top of page 234 – a piecewise arrangement.

So this replacement is mostly a trick to make it work.

For instance – think about breaking up

y = |x| into piecewise.

It looks like this

Homeschool Math Algebra II with Trig Absolute Value To write the function, we have one part for x >= 0 and then one part for x<0.

To find this we let |x| = x when x>=0 and then |x| = -x when x<0. This swaps the signs for you – a trick. Try it by making a table of x and y for value positive an negative of y = |x|.
So again – this is more a trick to break it up than a FACT. See in the example right next the the word SOLUTION is says we are trying to break this up into a form that does NOT involve |x|.

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Why McGraw-Hill Over Prentice Hall for Algebra II with Trigonometry?

Question from Kathryn: 

Is there a reason why you use McGraw-Hill over the Prentice Hall Algebra II & Trigonometry? We’re coming from Jacobs Algebra I & Geometry and I want to keep the same teaching style for my son.

Answer from Dr. Callahan:

Our view on textbooks is not to keep the same teaching style, but instead to prepare students for college and life. Therefore, our goal was to find popular college level textbooks. See
http://www.askdrcallahan.com/pages/The-Math-Problem.html
for more information on this thought.

While I am sure that having the same teaching style over and over is great for short term learning – it actually limits the students ability in the long term.

Hope this helps.

dwc

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Ready for Algebra II with Trig after Geometry?

Question from Diane:

My son is using your Geometry course and is doing very well. Geometry seems to be an easier subject for him than Algebra was. I want to use the next course, Algebra 2 with Trig, but I am concerned because he had a difficult time with Algebra 1 (even with the dvd’s). What do you suggest?

Answer from Dr. Callahan:

Put extra focus on those algebra reviews in the geometry book. If he struggles with some of them, take time to review the subject. He is getting more algebra than you think doing geometry anyway – but that review should help lock it in. I might have him go back and do a final test (again) in algebra when he completes geometry to see if he is still weak in areas. It would be worth it spending time reviewing algebra again after geometry if he has not gotten it yet. When done – let us know how it looks and we can suggest some specifics.

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Algebra II with Trig Ch4.2

Question from Rachel:

I’m on 4.2 of Algebra 2 at the moment.  I made it through the lesson and understood it until I got to the last section, titled “Finding Rational and Imaginary Zeros.” I understood most of that with the exception of a couple of things. Here are my questions.

#1) Why, when using synthetic division to find the zeros of the polynomial does the book only use the positive version of the possible numbers? For example, +/-1 and +/-5 were the possible rational zeros in Example 5. In the division table, they only used a positive 1. I know that was because they found the zero right off, but from what they’ve been doing, they would only have used positive 1 and positive 5. Could you tell me why?

#2) In the matched problem after example 5, the polynomial looks like this:
P(x) = X^4 + 4X^3 + 10X^2 + 12X + 5.

When you find the possible rational zeros (+/- 1 and +/- 5), and plug them into the synthetic division table, none of them (as positive numbers – like the book has been using) turn out to be zeros!
On the 5 for example, 5 times 1 is 5. Added to 4, it becomes 9. 9 times 5 is 40. Added to 10 it becomes 50. 5 times 50 is 250. Added to 12, it becomes 262. 5 times 262 is 1310. Added to 5, it is 1315.

Now I know that that’s not supposed to be happening. I just don’t know why it is! It seems like there needs to be some negatives somewhere to cancel things out. I could use some help figuring it out.

Thanks for all your help! I appreciate it!

Answer from Dr. Callahan:

The book only used the positive numbers in Ex 5 because they just started there and they happened to work. Nothing else, just they started with the easiest.

In the other, 5 does not work because it is not a zero. Just because you have POSSIBLE zeros, does not mean they work. You are testing values to see which work and which do not. When they do, you have factored it down.

See http://www.wolframalpha.com/input/?i=factor+X%5E4+%2B+4X%5E3+%2B+10X%5E2+%2B+12X+%2B+5

This shows that you have ONE zero here at x=-1. Also, note that you cannot factor anymore since there are no other REAL zeros.

Remember, the whole thing here is just a tool to find placed where the equation =0.

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Algebra II with Trig vs Precalc

We get a lot of questions about the difference between Algebra II with Trig and Precalculus.

Much of this comes from the publishers themselves. Here is my view…
1. Algebra II with Trig and Precalc are usually the same things – very little difference. To see this you could buy the Alg II with Trig book and the Precalc book by the same author/publisher and put side by side. Almost identical. Why do this? Marketing. If a school requests precalc, you want to have something to offer. If Alg II with trig, same thing. Do not miss a sale because of a cover. While you might find some texts that differ, I would not get caught in the mud worrying about it. See below for ACT/SAT.
2. If one takes Algebra II only, then you need some trig. Could do a one semester in trig using our book, websites, or something like a Trig for Dummies book. As long as you get the basics of trig formulas, simplification of trig identities, the law of sines and cosines, and some practical uses, you are good. For those who have had some algebra II, we direct to the quizzes for our text. found here http://www.mhhe.com/math/precalc/barnettcat7/ and have them work quizzes from chapter 1-5. If they can do this with 80% or better, they have the Alg II portion and can now move to trig. If they can do well on chapter 6-8, they have trig. (Again – about 80%)
3. ACT and SAT are key. I am not saying teach to the test – but yes, teach to the test. These exams are written so student MUST understand the material to do well – at least in the math part. This is all you need for college and for $ for college (if anyone cares;) The big problem in college calculus is a lack of understanding of basic algebra. Students often can do the work if they see examples of like problems, but given something new, they do not know the concepts enough to make it work. So, they flunk out of college calculus.
4. Cookbook math – related to ACT/SAT we often tell people math is like cooking. I can cook a decent meal with a cookbook if I follow instructions. Hardly makes me a chef since I do not understand how ingredients go together. A true cook could be let loose with ingredients and make-up a meal. They understand how things work. Math is like this. Too often we teach cookbook math, meaning the student can only solve a problem if formulated in the steps they have seen before. But, if thrown a real-world problem, they are lost. I realize this is not so much about Alg II etc – but this is a HUGE issue we see in science and engineering. We have a huge demand for people to do this kind of work, but most cannot hack it because they are cookbook math people.
More details in The Math Problem.
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A2T Ch 5, Lesson 5 #37

Start by taking log of both sides to get

Log ( x^Logx) = Log (100x)

now using rules from page 380

bring down the exponent in front and separate the multiplication into addition to get

Logx (Logx) = Log 100 + Log x

and Log100 = 2

now we have

(Logx)^2 – Log x – 2 = 0

factor to get

(Logx -2) (Logx +1) =0

now you have two equations to solve for 0.