Math Moment with Divi: Counting the Stars

For a long time, I’ve dreamed of using video as a vehicle to communicate about God’s handiwork in math. Below is my first attempt: a 35-second video in which Divi (a division sign) sets out to count the stars…and realizes just how much bigger God is than we could ever imagine.

Please let me know what you think…and feel free to share the video with a friend.

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Make It Real Learning Activity Library

Make It Real Learning Library

11/28/2012 Update: Make It Real Learning now has a Volume 2 as well. This post was originally a review of Volume 1, but I have added information on Volume 2, as they are very similar. Look at the bottom of the post for a 25%-off coupon good at Make It Real Learning through the end of December 2012!

I recently had the opportunity to review the Make It Real Learning Activity Library–a collection of e-books filled with practical worksheets that truly give students the chance to use math in real-life scenarios. While they do not come from a biblical worldview, their format lends itself to the parent picking and choosing which scenarios to use as well as discussing them further and could be a resource for those wishing to bring in practical examples.

I’ve put my entire review below. If you’ve used the product, please feel free to leave your thoughts in a comment!

Review of  Make It Real Learning Activity Library

Publisher: Make It Real Learning
Grade Level(s): K-College (Volume 1)/K-Grade 3 (Volume 2); see first paragraph below.
Price: $39.99/each volume of e-books (11 in each volume). Note: The publishers have offered a 25% discount (good through the end of 2012) for readers of this blog; this is not an affiliate code, and I do not get any commission on it. I am just passing it along in case any of you wanted to use it. To use the discount, enter christianperspective25 as the code during checkout at www.makeitreallearning.com.
Where to Obtain: www.makeitreallearning.com

Like its name implies, the Make It Real Learning series by Frank C. Wilson seeks to make math real for students, answering the question of “when am I ever going to use this?” Each volume in the series consists of 11 e-books, each one of which contains 10 real-world scenarios. The e-books range from one on fractions, percents, and decimals to e-books on more advanced topics such as geometry, algebra, linear functions, and quadratic functions. The majority of the e-books deal with upper-level concepts. The website offers a general mapping of lessons to grades for volume 2, although homeschool parents should be aware that most homeschoolers are not required to follow these standards* (many homeschool curriculums vary from them) and thus will want to look at the lessons to see if their student knows the needed skills.

Each real-world scenario stands on its own and can be printed and handed straight to the student. Duplicate worksheets containing answers (and often detailed solutions) are included. The formatting is professional and clean. The scenarios could be used as periodic assignments to both provide a refreshing break from everyday math lessons and to teach students to use math practically. The e-books do not typically present any of the math itself (so you will want to make sure your child knows the information needed to complete the scenario); they are designed for the student to apply what he has learned or is learning to real-life scenarios.

The scenarios themselves vary greatly. Some of them rank among the most excellent, well-thought-out activities I have encountered. For example, students will get to find the cost of keeping a pet (using real petsmart.com data), make cell phone comparisons and investment decisions, examine different pool designs, and understand the math behind various pieces of data all around us we take for granted. On the flip side, the books also include scenarios mentioning topics I found unnecessary, such as AIDS and teen pregnancy. I would plan on finding some great scenarios, but know that you also might find some you would not want to use or would want to discuss. Many of the topics, such as those on health or population issues, warrant discussions and explorations of a biblical worldview of that topic. Others, such as those that examine aspects of God’s creation (such as the phases of the moon or the order in sound waves) just need a reminder that God is the One who put this incredible universe together. Since the material comes as an e-book, you have the ability to select just the scenarios that will work for your family by screening them on the computer and printing only those you want when you want them.

The thing I loved about many of the scenarios is that, unlike a typical word problem, they really take the student into the scenario and let them experience the decision in a way few math books even approach. When used selectively, I can see them being wonderful ways to present math as a practical tool, especially in the high school years where textbooks focus more and more on abstract math.

25%-Off Coupon Code

The publishers have offered a 25% discount (good through the end of 2012) for readers of this blog; this is not an affiliate code, and I do not get any commission on it. I am just passing it along in case any of you wanted to use it. To use the discount, enter christianperspective25 as the code during checkout at www.makeitreallearning.com.

Disclosure: I requested and received a free copy of this product to review. See my review policy here.

* For information on homeschool laws by state, see www.HSLDA.org. This is not meant to be legal advice. Requirements vary state by state.

Free Videos: For All Practical Purposes

For All Practical Purposes

For All Practical PurposesFirst of all, thank you to everyone who provided feedback on the cover! It was VERY helpful. The graphic designer was able to make a few changes based on the concerns raised, making what I believe will be both a catchy and meaningful cover (I love the final design). Now it’s time to make some updates to the inside material and get it off to the printer : )

Secondly, I’d like to let you all know about a free video series that’s available on Google Videos: For All Practical Purposes. This series of 26 half-hour episodes does an excellent job presenting math’s practical uses in a fun and meaningful way. I was blessed by watching them several years ago when I was first beginning my  research, but I didn’t realize they were still available online until a website reader e-mailed me this past week with the news. (Thank you, Angela!)

A far cry from a boring classroom presentation, these videos make math both interesting and exciting through real-life examples and footage. I loved how the series made complex concepts simple, enabling the viewer to learn without even realizing it. One or two of the videos have very brief sections that discuss evolution from a non-biblical perspective, but on the whole the videos stayed clear of the topic of origins and focused on math’s practical uses. Since these videos were produced in the 1980s, a few videos feature rather archaic computers; however, the principles the videos present about math in action have not really changed.

This series is great for high-school students (or younger with assistance). One idea would be to watch a video a week as a supplement to your middle school or high school math course as a way of showing math’s usefulness in real-life situations–a usefulness that’s only possible because our consistent, faithful God holds all things together! The company that made this series has also produced a full sized high-school/college textbook by the same title. I was able to purchase one through AbeBooks (http://www.abebooks.com) for $3.99, including shipping.

Free Geometry Resource

Cornerstone Curriculum, publishers of the Making Math Meaningful curriculum series, is offering a rough draft of the first several hundred pages of their geometry course for free online. Based on a quick look at the course, it seemed to present geometry as a useful tool. I have looked at some of the author’s other resources and know he strives to help students really understand the concepts he presents and not merely memorize formulas.

Here’s a quote from one of the opening pages:

Geometry is all about measuring lines, angles, surfaces, solids, velocities and their interrelationships. In this study, you will act as a consultant, designer-planner, and builder. The projects will range from designing a tree fort in your back yard to planning the construction of a sidewalk and home on the hilly streets of San Francisco to charting the path of the earth around the sun. In the process you will learn the principles as well as the vast usage of geometry in everyday life. Geometry is used by graphic animators, artists, photographers, interior designers, engineers, architects, builders, construction teams, surveyors and doctors just to name a few.

The draft copy online does not contain answers to the problems, nor is it an entire course, but you could certainly use some of the application ideas or concept presentations from the free download. If you do, I’d love to hear how you liked it–as I’m sure would the author.

Algebra and Statistics Resources

While browsing the Internet today, I came across some fascinating videos that connected upper-level math concepts with real-life applications in an engaging, easy-to-understand way. The videos explore such varied examples as making fireworks and oil production–along with MUCH more!

The series are secular series, and I do not agree with some of the examples chosen and ideas presented in the videos, but they do contain very clear, helpful examples of math in action if you discern through some of the conclusions. It might be wise to discuss them together with your students afterward, and explore together what the Bible says about the various topics (the environment, health, etc.)

If you’re interested in the videos, I would suggest watching them soon, as the site dropped another video series on math they used to have, and purchasing the DVDs are VERY expensive. I think they rotate the content periodically.

Anyway, here are the links!

Algebra in Simplest Terms
Against All Odds: Inside Statistics

Bridge-Building Resources (All Grades)

In my previous post on shapes, we briefly talked about how learning shapes doesn’t have to be confined to a textbook–how shapes help us understand and appreciate the shapes God placed around us.

Understanding how shapes respond to pressure–as well as lots of other math concepts–plays an important role in building bridges. Here are two bridge-building resources you could use with your children as a way to teach them to use math as a God-given, real-life tool.

Golden Gate Bridge – This section of the Golden Gate Bridge site offers lots of useful bridge-making links. There are links you could use with younger children, as well as ones for high schoolers.

Build a Bridge – While this interactive page doesn’t get into much of the math behind building bridges, it gives students an opportunity to explore the properties of different bridge designs and determine which design would be best suited for various situations.

Thoughts on Pi

Someone recently wrote and asked me if I had any information on pi from a Christian perspective I could share. So here are some thoughts on this mind-boggling–and incredibly useful–number.

What Is Pi?

Pi, symbolized π, is “A transcendental number, approximately 3.14159, representing the ratio of the circumference to the diameter of a circle and appearing as a constant in a wide range of mathematical problems.”[1]

As this definition explains, pi is a transcendental–a number that keeps going on and on. To get a better feel, take a quick look at the first 100,000 digits of pi–it’s a mind-boggling number!

Numbers such as pi defy our comprehension. As I mention in Revealing Arithmetic when looking at different types of numbers,

“The infinite nature of numbers reminds us of our limited knowledge. As James D. Nickel points out, ‘The infinite nature of the natural numbers has a way of telling man’s reason, ‘Under certain conditions, you can never know everything there is to know about me.'” Although our understanding is finite, God’s understanding is infinite. Psalm 147:5 tells us, ‘Great is our Lord, and of great power; his understanding is infinite.’ How foolish it would be not to trust Him!”[2]

When we look at pi, our minds should turn in awe and wonder at God’s greatness! Sadly, though, there “is almost a cultlike following that has arisen over the concept of π.”[3] It’s a reminder of Romans 1:20-23:

“For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead; so that they are without excuse: Because that, when they knew God, they glorified him not as God, neither were thankful; but became vain in their imaginations, and their foolish heart was darkened. Professing themselves to be wise, they became fools, And changed the glory of the uncorruptible God into an image made like to corruptible man, and to birds, and fourfooted beasts, and creeping things.”

Has Pi Always Been Expressed the Way It Is Today?

Hardly! The symbol π is just a symbol man chose to help express that real-life ratio–the symbol has actually been used to mean other things! Like most math symbols, it has been adopted within the last several hundred years [4].

Throughout history, men have tried to more precisely define pi. I found it fascinating to read in π: A Biography of the World’s Most Mysterious Number that, when you really dig into the text, it appears the Bible accurately uses pi to the fourth decimal place (1 Kings 7:23)the book concluded that ”such accuracy is quite astonishing for ancient times.” [5] Not surprising considering the Bible’s Author!

Where Do We Use Pi?

Pi has a way of showing up all over the place–a testimony to the same Creator holding all things together. The most obvious use is when dealing with circles (for example, pi is used to find the area and circumference of a circle), but pi also proves useful in less-obvious places, such as in sound waves, general relativity, movements of the heavens, and probability, to name a few. 2018 Update: NASA has put together a Pi Day Challenge that shows just how useful pi is! Note that NASA does not come from a biblical worldview, so please use discernment (while most problems looked great, one at least hinted at finding life on other planets).

Why Are We Able to Explore Pi?

Because God created man in His image and gave him the ability to explore His creation! We’re thus accountable to Him for how we use that ability.

Where Can I Learn More?

There are lots of materials online that share more about pi (you might start with Wikipedia’s overview or this historical overview of pi). Your library may also have some books that could prove helpful. One I particularly enjoyed is π: A Biography of the World’s Most Mysterious Number by Alfred S. Posamentier and Ingmar Lehmanne.

The thing to keep in mind is to turn the wonder at pi itself into wonder at the Creator of all things–the one who understands what only baffles our comprehension.

Additional Thoughts (added March 2017)

Be careful when exploring pi to stand in awe of the Creator and not the creation itself. Throughout the years, men have been in awe of the number pi…yet that awe has often turned into an “almost cultlike following”[6] to the number instead of into awe of the Creator. To help put things in perspective, here’s a brief excerpt from Principles of Mathematics that explores pi, how it points us to the Creator, and how it’s been sadly twisted. http://www.christianperspective.net/wp-content/uploads/pi.pdf

 


[1] The American Heritage Dictionary of the English Language, New College Edition, 1980, s.v. “pi.”

[2] Loop, Katherine, Revealing Arithmetic: Math Concepts from a Biblical Worldview (Fairfax, VA: Christian Perspective, 2009), p. 125. Internal quote from James D. Nickel, Rudiments of Arithmetic: Foundational Principles in the Computation and Theory of Numbers, 1st ed. (preliminary draft) (U.S.: James D. Nickel, 2008), p. 294.

[3] Posamentier, Alfred S. and Ingmar Lehmann, π: A Biography of the World’s Most Mysterious Number (Amherst, New York: Prometheus Books, 2004), p. 13.

[4] It was “introduced in 1706.” Cajori, Florian, A History of Mathematical Notations: Two Volumes Bound As One (Mineola, NY: Dover Publications, 1993), vol. 2, p. 9.

[5] Posamentier, Alfred S. and Ingmar Lehmann, π: A Biography of the World’s Most Mysterious Number (Amherst, New York: Prometheus Books, 2004), p. 28.

[6] “There is almost a cult like following that has arisen over the concept of π.” Ibid., p. 13.

Thoughts on Shapes

Throughout history, men have used their knowledge of shapes to help them design buildings. Because of the consistent way God holds things together, we can predict how different shapes will hold up under pressure. One shape that supports weight well is the arch. You can easily see this by holding a piece of paper flat between your hands and having someone push down gently in the center of the paper. You should notice the paper bends easily under the pressure. But if you bend the paper to form an arch, you’ll notice the paper does not bend as easily. An arch shape holds up better under pressure than a flat shape does. [1]

Knowing this quality about arches helps us in designing buildings and bridges, many of which have an arch shape! It also gives us new appreciation for the design in our feet. If you run your finger along the bottom of your foot, you will feel multiple arches on your foot! God, the master engineer, designed the shape of our feet to support our body’s weight. Our feet are truly marvels of engineering!

If the foot were flat and rigid, fixed at right angles to the bone of the leg, walking would be difficult or impossible. The elastic arches also serve as shock absorbers to soften the jar resulting from walking on a hard surface.

The human foot is a miniature suspension bridge which is much more complicated than an ordinary bridge. Would anyone say that the Golden Gate suspension bridge just happened? Of course not, if he were truthful! But why do people assume that the even more intricate mechanism of the human foot could have just happened without intelligent cause or the workmanship of a master Engineer? [Allen L. Gillen, Body by Design (Green Forest, AZ: Master Books, 2001), pp. 43-44.]

The point? Learning about shapes doesn’t have to be confined to a textbook! As you teach your child shapes, you can be teaching him about the shapes all around us–and seeing the Creator’s wisdom and care in how He chose just the right shape for everything.


[1] This experiment is based on one given in The Art of Construction. The book offers numerous experiments and information related to building. Mario Salvadori, The Art of Construction: Projects and Principles for Beginning Engineers and Architects, 3rd ed. (Chicago: Chicago Review Press, 1990). Originally published as Building: The Fight Against Gravity.

Lessons from the Life of Johannes Kepler

Johannes KeplerKnown as the discoverer of the laws of planetary motion, Johannes Kepler was the first to propose that the planets circle the sun in elliptical shapes rather than in circular shapes as previously thought. Although often thought of as a scientist, Kepler was also a mathematician. In his study of planetary motion, Kepler used extensive math, including definitions, geometry, trigonometry, algebra, and other math concepts.

This mathematician’s life both provides an example of math in action and is resplendent with lessons! Join me in taking a brief look.

God’s Plans Are Not Always Ours
Johannes Kepler’s life illustrates the important truth that God’s plans are much better than are own–and sometimes they surprise us! Johannes Kepler did not plan on becoming a mathematician–he set out to become a minister. But toward the end of his university studies, his professors recommended him for a math position.

The young minister-to-be didn’t like the idea of giving up his divinity studies. Although he eventually agreed to take the position, Kepler still planned on becoming a minister one day. But God had something very different in mind for Kepler, as Kepler himself later recognized. [1]

Kepler had always been interested in the movement of the heavens and had admired Copernicus and his sun-centered theory. As a professor, Kepler now had more time to investigate these matters. He spent years developing a theory to explain the movements of the heavens, only to later discover his theory was insufficient. Undaunted, Kepler kept trying. His belief in the universe as an orderly creation of God made him certain the movement of the heavens could be explained by geometry. [2]

In 1600, Kepler’s teaching career at the school came to an abrupt halt. Along with others who refused to convert to Catholicism, Kepler was told to leave the country! Yet although Kepler probably could not see it at the time, God had a plan to transform persecution and exile into a tremendous blessing.

BraheExiled from his own country, Kepler soon found himself assisting (and depending on the generosity of) the famous astronomer Tycho Brahe (pictured to the left)–that is, until Brahe died in 1601. After Brahe’s death, Kepler inherited Brahe’s position and records. Because he had been exiled from his own land and forced to take shelter under Tycho Brahe, Kepler now had the records he needed to discover the laws by which God caused the planets to orbit the sun. Who would have thought God would use a persecution and forced exile to help Kepler accomplish his life work?

Perseverance in the Face of Obstacles
One of the biggest lessons we can learn from Kepler is that of perseverance. Discovering the planetary laws did not prove an easy task. From a collection of numbers Brahe had made over a period of many years chronicling where in the sky Mars had appeared, Kepler tried to find some sort of orderly law that could express the way God caused Mars to orbit the sun.

Just how hard was this task? According to Robert Wilson, “It took Kepler eight years and nearly a thousand pages of closely written calculations before he cracked the problem and discovered his first two laws of planetary motion (the third was to wait another nine years).”[3] Can you imagine spending eight years on a geometry problem you are not even sure can be solved, then another nine years to finish the task?

Confidence God Created an Orderly Universe and Math Could Describe It
Kepler’s willingness to persevere came from his deep faith that God had created an orderly universe. Kepler longed to uncover that order so he might bring glory to His Creator and know Him better.

Kepler was unwilling to accept the “close” results obtained from the prevailing Greek cosmology of the universe, in which planets circled the sun in circles. Instead, he searched for a better model.

Questioning the Greeks was a huge step. For centuries, the Greek philosophers’ teachings had been taught as fact. To question them was equivalent to questioning proven fact. Kepler could only be so daring because he believed in God as the source of truth, not the Greeks’ human reasoning. [4]

Kepler’s Beliefs – The Good and Bad
No one reading through Kepler’s Harmonies of the World can doubt Kepler’s belief in God. He often paused in the middle of an explanation to mention his Creator, and sometimes even broke off into a hymn of praise. It seems almost as if Kepler still viewed himself as a minister, trying to uncover the glory of God throughout creation. His book on planetary motion ends with this tribute to God:

Crying out with the royal Psalmist: Great is our Lord and great His virtue and of His wisdom there is no number: praise Him, ye heavens, praise Him, ye sun, moon, and planets, use every sense for perceiving, every tongue for declaring your Creator…To Him be praise, honour, and glory, wourld without end. Amen. [5]

We can learn a lot from Kepler’s use of math as a tool to uncover God’s handiwork.

At the same time, though, Kepler’s theology and outlook on math were far from perfect. He carried over a lot of Greek mysticism into his beliefs about God and the universe. Forgetting that creation and our minds are both fallen, Kepler often drew unbiblical spiritual parallels and inferences about God. Kepler also dabbled in astrology (although he admitted it held no weight) and brought a good deal of mystical thinking into his astronomy.

Lesson? We need to be on guard against falsehoods and lies from our culture that try to creep into our hearts.

Conclusion
Within Kepler’s life, we see God’s sovereignty at work, using even an exile to accomplish His purposes. We also find a challenge to persevere–and to view the universe as God’s handiwork and worship Him while using math to explore it. At the same time, we find a warning to be careful about falsehoods that threaten to rob us of living in the completeness of God’s truth.


[1] Max Casper, one of Kepler’s biographers, says, “Looking back later when, through the discovery of his planet laws, he had become aware of his ability, he recognized the voice of God in the call which had come to him. It is God who by a combination of circumstances secretly guides man to the various arts and sciences and endows him with the sure consciousness that he is not only a part of the creation but also partakes in the divine providence.” Max Caspar, Kepler, trans./ed. by C. Doris Hellman (New York: Dover Publications, 1993), p. 51.

[2] Kepler believed God was geometry’s creator. “For the Creator, who is the very source of geometry and, as Plato wrote, ‘practices eternal geometry,’ does not stray from his own archetype.” Kepler, Johannes, Harmonies of the World, in On the Shoulders of Giants: The Great Works of Physics and Astronomy ed. Stephen Hawking (Philadelphia: Running Press Book Publishers, 2002), p. 645. The Bible, however, never tells us God “practices eternal geometry.” We should not be surprised to find that parts of God’s creation are even more complex than geometry can describe. Nonetheless, Kepler was right in his general belief that God created an orderly universe, and that math records that order.

[3] Robert Wilson, Astronomy Through the Ages: The Story of the Human Attempt to Understand the Universe (Princeton, NJ: Princeton University Press,   1997), p. 69. For more details about the obstacles Kepler faced, see Max Caspar’s Kepler or Robert Wilson’s Astronomy Through the Ages: The Story of the Human Attempt to Understand the Universe.

[4] See Chapter 6 of Beyond Numbers for a basic overview of the switch from Greek thinking to biblical thinking that paved the way for the Scientific Reformation.

[5] Kepler, Johannes, Harmonies of the World, in On the Shoulders of Giants: The Great Works of Physics and Astronomy ed. Stephen Hawking (Philadelphia: Running Press Book Publishers, 2002), p. 723.

——
Resources Consulted

Caspar, Max. Kepler. Trans./ed. by C. Doris Hellman. New York: Dover Publications, 1993.

Kepler, Johannes, Harmonies of the World, in On the Shoulders of Giants: The Great Works of Physics and Astronomy. Ed. Stephen Hawking. Philadelphia: Running Press Book Publishers, 2002.

Newman, James R., ed. The World of Mathematics. Vol. 1. New York: Simon and Schuster, 1956.

Nickel, James D. Mathematics: Is God Silent? Rev. ed. Vallecito, CA: Ross House Books, 2001.

Tiner, John Hudson. Champions of Mathematics. Green Forest, AR: Master Books, 2000.

Wilson, Robert. Astronomy Through the Ages: The Story of the Human Attempt to Understand the Universe. Princeton, NJ: Princeton University Press, 1997.

Musings on Algebra

For our first specific concept post (see the schedule), I thought I’d offer some general thoughts on a part of math that confused me for years: algebra. Hopefully, this will help you see God’s handiwork amid the xs and ys.

What is algebra?
Algebra is the process of using letters and symbols to describe general quantities and relationships.

For example, say I head into a store with $20 and come out with $5. I would have spent $15.

$20 – $5 = $15.

My starting dollars minus my ending dollars showed me how much I spent in the store.

starting dollars – ending dollars = dollars spent

Let’s use letters to represent this relationship. We’ll use z to represent the starting dollars, y for our ending dollars, and x for the dollars spent.

z – y = x

We now have represented a general relationship that holds true for more than one situation! (We could use it to see how much we’d spent in any store.)

The above is one example of using letters to describe a relationship. By recording a relationship rather than a specific situation, we’re able to solve for unknowns and discover other relationships. This process proves useful in MANY areas (electricity, interest rates, gravity, laws of motion, etc.)

Algebra is based on the idea that certain relationships consistently hold true–that objects operate in a predictable fashion. Dollars do not mysteriously multiply in our wallets. Objects fall in a predictable way. The relationship between the power and voltage and current in an outlet remains the same. Why do things operate so predictably?

Because an unchanging God holds every aspect of this universe together! If God were not keeping everything together in an amazingly predictable manner, algebra would be completely useless. But because of God’s unchanging hand over creation, we can use letters and symbols to name and describe the predictable world around us.

What about all the rules?
A large portion of algebra textbooks focus on rules and conventions. Each “rule” is one standardized convention to represent a real-life consistency.

For years men did not use our current conventions at all! The graphic shows some different ways an algebraic equation has been expressed.

Why does algebra often seem so meaningless?
So often, algebra students completely miss seeing the amazing consistency algebra records because they get lost in the mechanics. As Morris Kline points out, “The usefulness of the techniques of algebra has caused many people to mistake the means for the end and to emphasize these menial techniques to the exclusion of the larger ideas and goals of mathematics. The students who are bored by the processes of algebra are more perceptive than those who have mistakenly identified algebraic processes with mathematics.” [Morris Kline, Mathematics and the Physical World (1959; repr. and slightly corrected, Mineola, NY: Dover Publications, 1981), p. 68.]

Conclusion
As you teach algebra, beware of emphasizing the means (i.e., the rules and conventions) to the point that your student loses site of algebra’s purpose–to record consistent relationships. Remember to let your mind pause and consider the greatness, power, and consistency of the God who, day in and day out, governs all things consistently enough for us to record general relationships and expect them to hold true in various situations. His power, might, and faithfulness truly know no bounds!

“I am the LORD: that is my name: and my glory will I not give to another, neither my praise to graven images.” Isaiah 42:8 (KJV)

Note: Watch for more on algebra soon!