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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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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)

h) Final Grade  (c + e +g)      ____82____(round up to nearest integer)

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The Different Geometry Textbooks and Videos

The 3rd edition of the Harold Jacobs Geometry textbook has gone through three different printings with 3 different covers and ISBNs. If you are picking a text and looking at used options, it can be confusing. So let me try to help.

First, all 3 versions are basically the same – as long as they say 3rd edition. They look like the photos below.

The Three Geometry Textbook Versions that Still Work

The original published by Freeman – ISBN: 978-0-7167-4361-3 and the corresponding Teachers Guide which contains the solutions to the problems. This book had a separate Test Bank for tests. The second printing published by My Father’s World – ISBN: 978-1-61999-109-5 and the corresponding Teachers Guide which contained the tests and the Answer Key.

The third printing published by Master Books – ISBN: 9787-1-68344-020-8. This is the latest and if you are buying new, this is what you are getting. The tests are in the Teacher’s Guide and the answers are in the Solutions Manual. You will need all 3 books for the course.

The Geometry Videos that Support the above Books

All 3 of the printings will work with the AskDrCallahan videos. The various printings have minor changes in page numbers, drawings, and some problems have changed – but the basic content remains the same. But, just to be more confusing, the videos come in 3 different packages. All are the same content but they might look different. All require the student to have one of the 3 textbooks, the tests, and a set of solutions.

1 – The online videos, offered by AskDrCallahan, contain the same instruction, and bypass the need for DVDs and a DVD player. They are the same content as on the DVD. The online videos come in two options.

• Monthly option – Pay for the course monthly and cancel anytime by logging into your AskDrCallahan account (or email us to cancel). This option is perfect if you think you will need just a few months.
• Lifetime access – If you need a full year, or have siblings who might use the course later, then this is your option. One price one time.

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I am used to having parents tell me that they think their kid is behind, missed something in 3rd grade, or has a suspected learning disability. That dialogue is almost always the first thing parents say when they are outlining why they need help from a math tutor. For some students, however, they do not need to be held back or to see a doctor. What they need is to move ahead and go on to the next step.

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MIT Makes videos for K-12 Students, and they are kind of awesome. (plus, unicorns!)

When I find great resources, I like to share them on the blog for parents and educators who might be looking for just that tool. I do not have any professional affiliation with this product, I just like it, and I want to share with you things that will help make your education experience the best it can be, with hopefully the least amount of obstacles for you.

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Innovation in Life Starts with Higher Level Math in High School

Are you panicked about Calculus? Maybe it’s time to be more panicked about a world where no one builds bionic arms, solves environmental problems, or seeks a cure for cancer–because that’s the reality you’ll have if kids stop taking higher math in high school.
Continue reading Innovation in Life Starts with Higher Level Math in High School

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Real World Math | The Algebra of Planning a Road Trip

When you plan a road trip, you use math literally every step of the way. What you may not know, however, is the right math term for what you’re calculating. So today, I’m going to walk you through a few steps of planning a road trip so you can see how Algebra helps you get from point A to point B successfully. Continue reading Real World Math | The Algebra of Planning a Road Trip

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3-step project idea to help you learn why learning math is important

Around the world, education is not as often available as it is in more developed countries. Too many high school students that I work with here in the US take math education for granted. You see, a global perspective on mathematics and it’s place in the world is a problem we, as the adults, created–or at least allowed. Here’s what I think we can do to change that narrative and start a conversation that creates a better perspective on math and education in general.

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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.