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## Mathematics in the World You Live In

It’s all connected. Math helps us understand better the world around us, from art to medicine to science. Math explains so much of it. Math has even been called the language of the universe. In short, God uses math all through His creation.  Therefore, it’s no surprise that humans use math in all they create as well.

In Algebra 1 we begin to understand ratios by starting out with fractions. We study spirals by understanding square roots.  Chapter 12 of our Algebra 1 course has especially fun square root illustrations with the horn of a ram for adding and subtracting square roots and the building of the Parthenon for dividing square roots. Here is a research article on how the Greeks used the Golden Ratio (often called the Divine Ratio) to build the Parthenon.  “This article will provide the best evidence I’ve found to date to illustrate appearances of golden ratios in the design of the Parthenon.” He has some really great illustrations in his article.

Art is used in our Algebra 1 Chapter 15  to show how the Golden Rectangle is used in famous artwork.  “The important relationship of mathematics to art cannot be understated when discussing Leonardo’s later work, and in numerous documents, letters and notes, the relevance of this is well documented. At times, he seems obsessed with these issues: while working on Mona Lisa for example, Leonardo is reported by Fra’ da Novellara to be concentrating intensely on geometry.”  – From this great article on Leonardo Da Vinci’s detailed math in his works of art as well as his scientific endeavors.

In Algebra 1 we learn how to manipulate numbers and use equations to find answers to real-life problems. Geometry takes those algebra skills and uses them to explain further how they can be used to draw, build, and even how math is abundant in nature.

Check out this fun article and see from hurricanes to dolphins to galaxies, we experience this amazing, and fittingly named, Divine Ratio.  Once we move into Algebra 2, Trigonometry, and Calculus we really see more fully the math come alive in the world around us.  How cool is it that we have the opportunity to understand it? Isn’t it awesome?

Let’s do some math! and never forget the big picture we’re learning toward.

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## Calculus Ch1.1 #44, 46, 48, 50

Here are the solutions to these even-numbered problems.

Calc1-Ch1.1#44-46-48-50.pdf (The PDF has #49 instead of #50. See below for #50)

Chapter 1.1 #50

What you are looking for is a function like
C = f(x) where C is cost and x is mile
If 0 <= x <= 1, C=2
If 1 < x < =2, C = 2 + 0.2(10(x-1))
Using Wolfram here is what it looks like

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## Scope and Sequence for College Bound or Not

Your cover school or state may require 2-4 high school maths. If you are working toward smaller scholarships or high-end private colleges, they may have requirements for specific courses in high school. Contact the colleges you are interested in and ask for their requirements. However, for the most part, colleges do not really care as long as they get money, and your student scores well enough on the ACT and SAT.  You also need to contact the college of your choice to find out the score required on these exams. Sometimes the front line admin people will say they want things – but really they want money. 😉 Know that no matter how high your student’s ACT/SAT score is, they will be required to take a math placement exam at the university and will be placed according to that score. Our courses will help with the ACT/SAT and math placement exams.

So as far as a typical math sequence goes, you can break it up in all kinds of ways. The key is how much time did you spend and what does your school (cover) require. We see some who call it

and others

Algebra
Geometry
Algebra II (do this course over a longer period)
Trig
Calculus

or

Algebra
Geometry
Algebra II (do first few chapters of Alg II – say chapter 1-4)
Precalc (Do 5-8)
Calculus

So it really just depends.

What I would suggest is to first find out where your student plans to go to college (or the options including what specific fields of study) then contact them and ask what they would like to see. Once you get to the department level advisors, you will get some good counsel. DEPARTMENT LEVEL is key!!! Not just the admissions or Freshman counselors.  I would do this no matter what. They might have other helpful hints you can use now. Do not worry about it being too soon. You are making a plan, and rather than listen to all the common advice about what they want to hear, just hear it from them.

But trust me, even in engineering we would never run off a student for lack of high school courses or high school grades. They are going to focus on those test scores and their own entrance/placement tests. (Which are written by the same people who write the ACT.)

## Not Going the College Path? Want some Math Application Studies?

Maybe you aren’t headed to the ACT/SAT or college admissions office. You just want to learn some useful, real-life math applications for entrepreneurship endeavors, trade careers, and controlling your financial life. Here is another scope and sequence that’s good education for everyone.

Algebra
Geometry
Personal Economics  (3rd-year math requirement)
Career Planning (NOT a math credit, but a great way to spend a summer or junior/senior year defining the right path for your student’s future) – a great business or elective credit.
Entrepreneurship 101 – Building Your Own Business (Not a math credit, but definitely a career path.) – a great business or elective credit.

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## Calculus 1 Course Description for Transcript

If you are looking for details for a transcript for Calculus 1,  this should do the trick.

Calculus is about change. The tools of calculus allow us to model change. Once we have equations for something, we can use calculus to see the impact of change on the system.

In Algebra, we learned how to generalized basic arithmetic to show relationships. So, instead of saying 2 + 2 = 4, we might say 2 + x = y, allowing a more general equation that shows a relationship between x and y.

In Geometry (which also includes trigonometry), we studied the relationships of shape.

In Calculus, we study continuous change. For example, our car gets so much gas milage which is impacted by the car’s weight. But the longer you drive, the less gas you have and therefore the less weight. So the weight of the car changes over time, therefore impacting the use of fuel over time.

Calculus is a necessary tool for any science or engineering. It is heavily used in physics and engineering and found in biology, business, social sciences, etc. No matter what we are studying, the only constant thing is change.

Calculus is broken up into two areas – Derivative and Integral Calculus. This course covers derivative calculus, which is equivalent to most Calculus 1 courses at most universities. Calculus 2 covers integral calculus.

Students who complete Calculus are prepared to enter college and should be able to step into a college-level Calculus 1 or Calculus 2 course – depending on how the individual university defines its course. Calculus is not required for the ACT and the SAT.

Topics Covered in this Course:

• Limits and limit laws
• Continuity
• Derivatives
• Differentiation rules
• Application of differentiation
• Antiderivatives
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## Geometry Introduction

The surfer/spotter problem of Geometry is answered on page 290 of the Enhanced Teachers Guide for Geometry by Jacobs (ISBN: 0716756072).  Since we no longer use this book, here are the two pages that solve the spotter problem.

FYI – We now use a smaller version called “Solutions Manual: Answers to Exercises for Geometry” which leaves out some of the old classroom teacher materials, including the above Sufer and Spotter solutions, but is cheaper and smaller. The older, larger version is no longer in print.

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## Calculators come in four various forms (as for math)

1. The 4 function calculator:  A few bucks and NOT what you want. These are: add, subtract, multiply, and divide.
2. Scientific calculator:  These calculators can do square roots, exponentials, and trig functions (plus many other things) and are great for high school math courses and can get you all the way to calculus – but not as good as the more powerful calculators. A scientific calculator can be found for a little over \$10. The TI-30 series is a great example. Most smartphones have this function built-in.  Turning an iPhone sideways will open up all these functions to the basic calculator. Be careful here as the touch screen is easy to punch in (or miss punching) incorrect information.
3. Graphing calculator:  These do everything a scientific calculator does, plus it allows you to graph functions. Start around \$50
4. CAS (Computer Algebraic Calculator) with graphing: These do all of the above plus computer algebra. In other words, they will solve problems for you. VERY powerful. They will typically be over \$100. the TI-89 is a popular example.

Any of the later 3 will work, but my choice for Algebra II with Trig or higher would be the TI-89 or equivalent. Not a ton of money for something you will use for a few years. A student could certainly do well with a TI-30 also, but if they will learn to use it, a TI-89 is a good deal.

The TI-30 Calculator is included in the Everything You Need Jacob’s Elementary Algebra and Jacob’s Geometry packages. This calculator is allowed on the ACT and can be used throughout your student’s academic career and college. You can use the TI-30 with Wolfram Alpha (see online calculators below) for your graphing needs if you don’t want to stick with the TI-30.

## Online Calculators

My favorite tool is the wolfram website.  It is far more powerful by far than any of the above. But, you cannot use it on any tests.

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## Geometry Ch3.6 #11 Solution

Question from Anna:

Geometry Chapter 3 Lesson 5 Number 11. How do we figure this out?

North East West South make 90-degree angles so <NOE = <EOS = <SOW = 90 degrees

A = 50. So to this point, draw with N = 0, A = 50 (given), E = 90, S = 180, W = 270

Now you have to figure out the rest. Here are a few ways.

1. You know <NOE = 50 = <DOS since these are linear pairs. So add 50 to the 180 to get 230 at point D. Keep doing the vertical angles.
2. Or you also know the lines of the map (north, east, south, west) bisect those angles < AOB and the like. So If <NOA = 50 and you know <NOE = 90 so <AOE = 40. And since they bisect, you now know <EOB=40, <BOS=50.

Either route you take, you just will see a pattern and you start figuring out one from the other.

So in the end the coordinates (angles) are 0,50,90, 130,180,230,270, and 310

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## Say YES to the calculator!

One of the most frequent math questions we get from homeschool parents is

“Do I let my students use a calculator?”

Is your student begging for the calculator? Have you stood firm against it all through elementary school? Do you feel like you are the “no” monster? Here’s your chance to provide a win without just giving in!

### Does Using a Calculator Mean You are Not Thinking?

Somewhere along the way, we got it into our heads that using a calculator prevented thinking. Those who pulled out the calculator were lazy and were never going to remember their multiplication tables.

While there might be some truth to that concern in the early years, once your students are in high school their math skills need to change from memorization to thinking – especially critical thinking. In courses like Algebra, Geometry, and above we are no longer trying to get them to memorize topics – but instead to understand concepts. And in understanding the concepts, the calculator can be our friend in two ways.

1. The calculator lets our mind focus on learning the concepts and not the labor of adding or multiplying numbers together.
2. The calculator actually makes us think harder. Whenever you plug in a series of numbers and get something you do not expect, like a negative number in a trig function, we have to think about why the calculator is giving us something we did not expect. In other words, the calculator helps us learn.

The truth is, the calculator cannot think for us.  In all reality, we can’t even use a calculator until we understand those basic mathematical functions of addition, subtraction, multiplication, and division. Otherwise, we’re just punching buttons.  We have to be able to know what an operation is before we can accurately punch that operation into the machine.

### What if My Student Has Not Memorized their Math Facts?

OK, a reality check is in order. If your student is in high school and still does not know all their math facts by memory – are you really going to hold them back?

Time to face it – if your student is at the high school level and hasn’t memorized those pesky math facts, they aren’t going to.

I will be honest here. I have a Ph.D.. in Engineering and I still need to think about it when I need to know  9 times 7. Judge me if you like, but I just never got it. Machines are made to help me.

The more skill your student has using a calculator, the better that tool works for them. They need to learn to use it to add and multiply so that they can later be comfortable with the tool when it is time to do more complex math.

So if they have not mastered the math facts – move forward anyway. You are holding them back. It’s a bit like continuing to study the alphabet when you’re actually ready to read Shakespeare, or chopping up cabbage by hand when you need so much done that a food processor is a smart way to go.

It’s time to move onward and upward! And the power of a calculator is essential to saving time so you can focus on critical thinking.

At the high school level (Algebra 1 and above), we are exercising and building critical thinking skills rather than rote operations skills. In elementary mathematics, our students were learning the mechanics of manipulating numbers. Long-hand calculations are helpful in learning what is happening in these operations. However, in higher math, we move from rote crunching to critical thinking and analysis.

### When a Calculator is Needed

We recommend using a calculator in Algebra 1 and up. Longhand is no longer needed. You will probably find a student new to the calculator using it when they don’t need to, such as for simple problems like 2×3. Don’t worry about it. Let them use it as much as they want, they will soon learn there are some operations that happen quicker in their heads than they can punch in.

They will also learn that they can’t always trust the calculator.  A calculator is a great tool and it only serves to strengthen what they have memorized (like looking at flashcards.) Using it early builds their calculator skills and experience in when to trust it.

You will find problems early in Algebra 1 that are simple addition, subtraction, multiplication, and division that are not intended as exercises in longhand mathematical operations. They are exercises in mathematical logic and understanding. These problems are highlighting relationships between numbers and the operations done on them. They are also great experience and skill-building in punching in operations correctly. The calculator only does what we tell it to. We must instruct it correctly and that takes practice.

When you get to Geometry, your student will find they use the calculator much less than they’d expect. Much of Geometry is logic. They will learn to think through situations, truths, and analysis. A little computation will come into play as well, but the bigger thinking is in the logic.

Algebra 2 with Trigonometry and Calculus will bring in the big guns of the calculator. Trigonometry will make the calculator a good friend of your student. Here they will find it tricky too, as again, the calculator only does what we tell it to, and if you punch it in wrong or in the wrong setting, you get a very wrong answer. You don’t want your student to wait until this stage to become familiar with a calculator. Start in Algebra 1 so they can develop the knowledge of the many operations the calculator can do as they are introduced throughout their high school career.

If you want to know which calculators are useful for which courses, check out this post on calculator types.

### Have Fun Exploring.

Learning can be a struggle, especially math. This is a chance to lighten the load without compromising the learning.

The calculator is part of the next step in your student’s learning journey. Let them be thrilled with your YES to the calculator. Let them overuse it. They will soon learn that the calculator doesn’t have all the answers. It only has the answers the student knows how to tell it to calculate.

High School math opens a whole new area to understanding God’s Creation. Have fun exploring and use the great tools that make that deeper understanding a better journey.

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There is no wrong way to grade. It’s completely up to you. You are in charge.

But sometimes we need some help deciding where to even begin. Our advice is to remember that the goal is learning, not grades. Grades are a reflection to OTHERS of how much you’ve learned. Most people can game this system and make great grades while knowing very little. This is especially true in math. Students are great at crunching the mechanics of math and can make the grade, but then actually have little understanding.

The ACT/SAT exams are geared to catch those deficiencies. They test for understanding rather than crunching. On those exams there isn’t enough time to crunch out the answers, there is a few seconds to guess the correct answer based on your understanding and elimination of impossible answers.

Math is critical thinking. Logic.

Learning and understanding is your goal.  At this point, your child is in high school – young adults. If they aren’t already, its time for them to own their education. Let them check their work. Help them understand, if they cheat, they are only cheating themselves.

Below are some options to choose from for grading. Using these methods, you may discover a combination of them or a completely different method works best for you. But here’s some ideas to start from:

Option 1:

With our children, we did not give a grade for daily work. We did have them check their work and redo the problems they missed.  The key here is to have the STUDENT check their OWN work. Then have them rework the missed problems – this is where learning happens!

Option 2:

Have the student check and rework their problems and give some credit, not for correct problems but, for the work being done as a percentage of their total grade.

Example:

30% of the total grade for homework (100% of this if they do and check all their problems)

70% of the total grade for tests

This option takes some pressure off the tests and incentive to do homework, however, if you follow our test grading guide that gives points back for corrections (again, where real learning happens), then the tests aren’t too pressured anyway.

We also give our tests as open book tests. Math books are good resources and everything in there can’t be memorized as you continue on into higher math. You will need the resource. If you are not comfortable with open-book testing, you could allow them a “cheat sheet” or “note sheet” where they put down the formulas, theorems, and anything from the sections the test is on to use during the test. Even in college math, we were allowed a cheat sheet of a certain size (usually a large notecard or half 8.5×11 paper) for formulas or anything we wanted to put on it.

The goal is to know how to use all the math tools, not to keep the toolbox in your head with no understanding of what they do.

In our AskDrCallahan Teacher’s Guide, you will find a test grading guide (mentioned above) that allows for regaining points for reworked problems.

Example:

Attempt # 1

a)  Number of problems correct ___30___

b) Total number of problems   ____50____

c) Grade  (100*a/b)            ____60____(round up to nearest integer)

Attempt #2

d)  Number of problems fixed  ____10___

e) Points added (70*d/b) ____14____(round up to nearest integer)

Attempt #3

f) Number of problems fixed  ____8____

g) Points added  (50*f/b)        ____8____(round up to nearest integer)