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Algebra Ch2.4 Set IV

Question from Cynthia:
Answer from Dr. Callahan:
You are correct. I assume you have a blue textbook and this is on page 72. It appears the problem got updated between versions and the solutions manual did not.
The y-axis is the weight of the contestant in pounds after he (she) eats each hot dog.
Then I can see in #2 that each hot dog eaten increases their weight in a direct variation.
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Raspberry Pi and A Robot Ferris Wheel you can build at home for less than $100

Some of the links in this article are affiliate links. We don’t carry in our shop all the materials you would need to complete this activity, so we use links to share with you the products you would want to use so that you can do this activity at home. Some of those links are affiliates and we will make a small percentage commission, at no additional cost to you, if you use our links to purchase your supplies.

Have you ever wanted to build your own Lego Ferris Wheel or Train and be able to control it like a robot? Now you can!–and BONUS it counts as a math class activity. Keep reading to find out how to do it and at the bottom I’ll show you where in your Algebra II with Trig course this activity would fall so you can schedule it in your lessons.

Continue reading Raspberry Pi and A Robot Ferris Wheel you can build at home for less than $100

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The Quadratic Equation

Where did all these formulas come from anyway?

All forms of math and science (including cooking, shopping, etc) are loaded with formulas. Formulas are nothing but a relationship between two different things.

The quadratic formula is one of the most well-known formulas in math – and one you will use again and again.

It is often interesting to see how formulas are derived. After all, it is very likely that you have, or will, have your own formulas you create for your hobbies or work.

Here is a great step-by-step explanation of how the quadratic formula is derived.

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The Homeschool Math Problem

The U.S. schools are weak in math – almost all of them regardless of public, private, or homeschool. The national weakness in math (and science – which is related) is a growing problem. Math and science are required for technological research, and research (like it or not) is required for national security. The fact is a majority of U.S. university graduate students in technology areas are from other countries – many with less than stable political systems. Congress and agencies that deal with national security are well aware of the problem – but fixing it is another challenge.

But what about homeschoolers?

We outperform everyone in everything – right? Wrong. Homeschoolers are weak in math.

“Homeschoolers need to do better in math. Our reading and language skills are excellent, even though we could always use a little improvement. But our math skills need real help. We only do slightly better than public schools here. We need to drill basic facts, teach concepts, and make sure we take our children through algebra II and geometry at a minimum.” [Mike Farris, “Aim high(er):,” World Magazine, April 28, 2001, Vol. 16. ]

Even though that quote is from years ago – not much has changed.

Besides the personal implications of future employment in a technology world, we need to consider the apologetic issues. Many Christians taken out of scientific debates about the origins of the universe, evolution, astronomy, etc due to their raw lack of basic knowledge about math and science. Remember – theologians of the past have been astute students of both the bible and nature – what systematic theology calls special and general revelation from God. (Note – our courses are not faith based – but we are Christians. We firmly believe that we do not need to force our faith into places like math – we just speak the truth. Many who have chased truth who do not believe in the God of the Bible have found the path to truth led them to their faith – including me.)

But the problem we have as parents is that we too were raised in a school system that was often inadequate to prepare us to teach our own children. Even then, most of us who did take the advanced math and science course have long since forgotten the skills we once had in the areas of math and science.

Are we preparing our children to live in an age of technology? …defend the faith against scientific attacks? …teach their children? Just as language is the way to study the written word of God, or special revelation, math is the language used to study nature, or general revelation.

So we have written this part to answer some other specific questions we often get about math.

What types of textbooks should I buy?

We suggest you lean toward a more college level in the Algebra II w/ Trig and Calculus courses – if not before. In fact, if you know where your child plans to go to college, find out what math they will need, get a syllabus, and use that text. (Our daughter would have to take one course in Calculus in her degree field, so we taught her the same calculus in the same book. She found the college calculus course just a review.) Note that the high school textbooks tend to be written with easier problems than the college level textbooks. See the From the Trenches below.

From the Trenches

As a fairly new member to the engineering faculty, I had learned that graduation rates in engineering and science had been down in all United States colleges and universities for the past twenty plus years. Most of this was because the high schools did not provide adequate training in math, so incoming students often got discouraged and moved into other fields. My curiosity drove me to call our math department and ask how students in general did in our calculus courses. The head of the undergraduate program explained that two local high schools outperformed all other students in math. (Both were public schools.) So I went to the math department at one of these schools and I asked what they did that made them better. The biggest issue was they used college level textbooks. They explained to me that the high school textbook publishers competed on how easy the problems were to work. The college level publishers would never survive if they watered down the material.
So the schools that use watered down material in their textbooks have built a large gulf between their math courses and the universities math courses. The few hours we spent at the local high school were convincing – and we have never turned back from college textbooks.

What about Saxon Math?
We often get asked about the Saxon math material. It seems people either love Saxon or hate it, but few are neutral. In our view Saxon provides an excellent base in the younger years when we are starting to learn the concepts of math as well as the basic facts. However, we steer away from Saxon once we hit algebra – and the higher you go the less we like it. Now before you send us letters of how well your children did in Saxon let us say that any math material can do a great job depending on the student.

Our problem with Saxon is not that it does not teach, but it does so in a choppy manner – or as Saxon refers to it – incremental. The incremental method is big selling point of Saxon – but it falls terribly short in the later grades for two reasons

  1. Saxon textbooks are difficult to use as a reference. Good math
    textbooks (algebra and above) should be good reference material for future math courses. Since Saxon does things incrementally, it is difficult to go and find a reasonable treatment of any subject in one place, therefore making Saxon material less than adequate for math reference. This also creates a challenge to parents trying to help their son or daughter with a concept. When we try to get to the root of an issue we often need to go back to where that subject was covered to make sure we (as the teacher) understand what the author is trying to do. In Saxon this is very difficult to do.
  2. Another problem with Saxon is it does little in the way of application. One thing we have learned from teaching math, science, and engineering courses is that application is very important. Often I am asked questions such as “why would anyone ever use this?” (I must agree, that is how I – and probably you – always viewed math.) When I hear this question I will discuss how these elements are used in engineering, sales and marketing, construction, medicine, and yes homemaking. We have had discussions about aircraft, electronics, lighting, the space shuttle, relativity, high blood pressure, lung capacity, cooking, household cleaners and chemicals, (on and on) as part of the answers to these questions. When these discussions take place with students I see an element of excitement in the students that was not seen before. Teaching someone what a hammer is and does is nice, but showing them how it is used to build a house is education! Saxon (and others) is weak in this area.

So, if you want to use Saxon, I encourage you to do two things. 1) Find a good math reference material. Some books such as Schaum’s outlines or related inexpensive material might suffice here. 2) Find a source to get at applications of the math. The best source of this material is to use the web, and have your children find ways in which the math they are using is used in the “real” world. However, this might be a challenge at times, so a supplement of another type of math text in the same subject would be useful. We will soon be offering a series of math experiments (expected to be about $25 per course) that will be usable with any math curriculum at the high school level. Join our email list to get updates on these supplements.

How does math fit into Classical Education?
Classical education has become very popular, and we are big fans of it. But frankly it is weak in math, and possibly weak in science. Conventional wisdom on classical homeschooling has the higher math courses as electives at best. Yet many sources for classical education recommend reading material written by Copernicus, Kepler, and Einstein as part of the science curriculum. Without knowledge of calculus, these works would be overwhelming. So if you like the classical approach we applaud you – just do not skip on the math! At the very least, you children should get through Algebra II with Trig.

What math material should I use?
Here is a big question! We have often been asked to review math material – and overall we are disappointed in what is available for homeschooling in the area of math. Not that the concepts are missing, but the method of presentation is similar to the presentation given in the public and private schools. Dry and without any application to the real world. College textbooks are (in general) richer in their treatment and application of the concepts. However, you may need some outside help with the textbooks since they are designed to be instructor based.

Any curriculum is OK as long as it meets the basic objectives of the course title. The rest is how well your child takes to it. If they love it, you have found a perfect match. The section Evaluating Math Texts will give some guidelines of picking the proper math curriculum.

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Calc1 Test 4 / Final Exam #6c

Beverly asked:

Looks like you have a simplification error in the second derivative. You have:

f”(x)=12x^2-4=4(x-1)
But it should be
f”(x)=12x^2-4=4(3x^2-1)
then the intervals for concavity all change.
Am I missing something? Thank you so much for your good product and wonderful support!
Dr. Callahan Answered:
You are correct. Thanks for pointing that out.
See the plot of the original equation (a good idea is to look at plots when things are not working out)
From here we can see how it is shaped.
Figure 1 – Original Equation
Now if we look at the second derivative, it should be
12 x^2 - 4
and look like
which has roots
x = -1/sqrt(3)
x = 1/sqrt(3)
and looks like
Figure 2 – 2nd Derivative
The concavity chart will now show up, down, and up but around the points
x = -1/sqrt(3)
x = 1/sqrt(3)
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Calculus Test 4 Sections 4.9-5.8 – #3, #8b, #9a, #9b, #10b questions

Question from Beverly:

#3 –  on the test copy I have says to use three equal subintervals.  But in the handwritten solution, it appears that they used 6. If the student uses 3 subintervals wouldn’t the ∆X be 2 and the right endpoint sum be 0.2, left end point be 39.8, and midpoint be 15.8?

Answer from Dr Callahan – That is correct – in the solution we used an interval of 1 and it should have been 2.

 

#8b – The handwritten solution has dV= √t and V = t^1.5.  Shouldn’t V = 2/3*t^1.5? With that, our final solution came out to be 3.39.

Answer from Dr Callahan – That is also correct – we have an error in our solution.

 

#9a – Instead of factoring the cos^5x into (cos^2x)(cos^2x)cosx, my student used the Form 74 from the Table of Integrals followed by Form 68. But he didn’t get the same answer as he would have solving it as we learned in 5.7 (p. 403) like the written solution did. Why don’t the two methods yield the same answer? I’m thinking it has to do with the limits of the integral, but it’s been 25 years since I had calculus, and I’m struggling with explaining the why on this one.

Answer from Dr Callahan – Yes you should get the same answer – but I noticed an error in our solution!!!! When you integrate we added a 4. Should be as below

NewImage.png

 

#9b – In the handwritten solution it looks like it evaluates 3*ln 3 as 9.89, but isn’t it 3.3? I believe the answer would be -0.17 if 3.3 is correct.

Answer from Dr Callahan –  Yes it should be 3.3

 

Extra Credit – The solution just substituted into the Form #95.  Did you not expect them to continue to solve the remaining integral or just stop with that first substitution?

Answer from Dr Callahan – Yes we just stopped there because that was the challenging part!

 

 

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Geometry Ch14 Test #4

 

Question: how do we work the problem on Chapter 14 test #4.

 

Answer from Harold Jacobs:

Since all 16 white regions are identical, we can find the area of one of them and multiply it by 16.

This figure represents one of the 16 small squares in the figure and one of the white regions.

 

2016 04 04 12 59 30

Since the diameter of the large circle in the figure shown on the test page is 4 units, the side of the square of one of the small squares is 1 unit.

We can find the area of region B by finding the area of regions B & C combined (a quarter of a circle) and subtracting the area of region C (a triangle):

> The area of B & C combined is1/4 (pi r squared) or 1/4 (pi 1 squared) = 1/4 pi

> The area of C is1/2 r squared or 1/2 1 squared = 1/2So B = (B + C) – C = 1/4 pi – 1/2
and the total area of the 16 white regions is16(1/4 pi – 1/2) = 4 pi – 8.