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.

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## 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.

## Balancing, Measuring, and Such

A few weeks ago, I visited a local historic home with my aunt, uncle, and cousins. Something I saw there sparked a little research and has resulted in this post : ).

The chandelier hanging in the hall of the historic house was much too high to light easily using a ladder. But men had used the ingenuity God had given them to design a device whereby the chandelier’s candle COULD be easily lit—by just one person.

The chandelier was hung using a long chain, at one end of which was a cylinder that perfectly balanced out the weight of the chandelier. Since the weight balanced, the chandelier didn’t move unless a person pulled or pushed on the chain. By pulling or pushing on the chain, a single person could easily higher or lower the chandelier, light it, and return it to its former position.

As I looked at the chandelier, I realized this was math in action. Whether or not the original designers weighted the chandelier and the cylinder, I don’t know. But I do know that the weights equaled—and that math has been historically used to help design and use counterbalances for a variety of purposes.

[Photo taken at Sully Historic Site in Chantilly, Virginia.]

Merchants used to use a scale based on the counterbalance principle to weigh their products. On one side of the balance, they’d place the item to be weighed. They’d then add items of which the weight was known to the other side until the scale balanced. The whole process required quite a bit of math.

If you’re working on measurement with your child, consider having him or her build a scale! Easy instructions can be found at the website below. As you build it, thank God for creating us capable of designing devices—like balances—to help us.

http://www.campsite24.ca/balance_scale_eng.pdf

## Money Fun

A recent trip reminded me of one practical way math serves as a useful tool–money conversion. On our trip, we saw prices listed in dollars, pounds, euros, forrents, and marks–and needed to use some quick math to figure out how much we were really spending : ). We employed addition, subtraction, division, and round/estimating at various times throughout the trip.

As you study different cultures in history, consider having your child learn about their money systems too. Simply search the Internet for “money” and the country’s name. You should be able to find some pictures and descriptions of the money in that country. Then search for exchange rates and see how the money compares to American money.

If your child is young and just getting used to American money and math facts, you could simply show your child the pictures of the money, explaining that different countries use different money systems just like they use different languages, and that math helps us compare prices. If your child is proficient with math, you could actually have your child pretend to go shopping in the foreign country and figure out how much something marked in that country’s currency would cost in American dollars. You could even set up a pretend shop!

Math is truly a useful tool!

## The Golden Ratio – The Creator’s Mark Throughout His Creation

Updated 11/13/15

Some time ago, someone wrote and suggested I write something on the golden ratio. I hope to finally post something on the topic today (this post has been in progress now for QUITE some time).

The “golden ratio” is a special name given to describe a ratio that seems to relate indirectly or directly to many aspects of God’s creation. The ratio is approximately 1.618 (see Wikipedia for a more exact definition).

To understand how we observe the ratio 1.618 in God’s creation, we need to take a look at a special sequence called the Fibonacci numbers. This special sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…and continues, with each new number formed by adding the previous two numbers together (1 + 2 = 3, 3 + 5 = 8, etc.). The ratio between most numbers in this sequence is very close to the golden ratio. This means that if you were to divide two neighboring numbers in the Fibonacci sequence, you’d get a number close to 1.618!

Since we find neighboring Fibonacci numbers all over creation, it follows that we also find the golden ratio all over too. For example, the seeds in any given sunflower are arranged in two patterns of spirals. If one of the patterns has 55 spirals; the other will have either 34 or 89—the number of spirals in each pattern are always neighboring numbers in the Fibonacci sequence! This also means that the ratio between the two patterns is always very close to the “golden” ratio. No matter how large or small the sunflower, one spiral pattern always contains approximately 1.618 times the number of seeds as the other pattern. Guess what? This ratio allows for the most number of seeds to fit in any given sunflower! God sure thinks of all the details, doesn’t He?

If we were to look at plants, pinecones, or pineapples, we would again find Fibonacci numbers and the golden ratio and be awed again at the Creator (please see the websites listed at the end of this article for a more detailed explanation). Scientists have even found ways where things like the nautilus’ shell relates indirectly to the golden ratio. And artists and architects have discovered that rectangles based on the golden ratio are artistically pleasing (surprise!).

Many marvel over how the golden ratio (and numbers in the Fibonacci sequence) keeps popping up all over creation. As Christians, we know the ratio appears everywhere because God designed the golden ratio to have the properties it has, and then designed each part of His creation with infinite care and wisdom, using this ratio to give sunflowers, pinecones, and more just what they needed. He also created our minds to appreciate this same ratio as something “beautiful”—as testified to by the many buildings and paintings that incorporate this ratio.

Below are four sites that offer more details about the golden ratio. The first two approach the ratio from a biblical perspective; the last two do not have a biblical perspective, but contain some fascinating information and easy-to-understand explanations of the Fibonacci sequence and the Golden Ratio you might find helpful.

Note that one of the sites uses the name “golden mean” instead of “golden ratio” to refer to this ratio. The sites also refer to the “golden rectangle” or “golden section”—this is a rectangle whose sides follow the proportion of the golden ratio (one side is approximately 1.618 times the other side). Most of the sites also discus the Fibonacci sequence—remember that neighboring numbers in this sequence have a ratio that approaches the golden ratio.

I hope you enjoy!

## Table Math

“How many math concepts could you teach using just the objects on this table?”

The question came from my brother. We had just finished eating dinner, and I was brainstorming with him about some decisions I needed to make regarding the layout and content of Unveiling Math

I looked at the objects on the table—our leftover food and an assortment of dirty dishes. These were hardly objects one would ordinarily think to use to teach math concepts.

Yet as I began to answer the question, I realized that one could really teach nearly every math concept using just the items on that table. Addition, a method of recording the way God causes objects to add together can be demonstrated by “adding” the forks or cups on the table! Subtraction, multiplication, and division could be taught in a similar fashion by subtracting, multiplying (adding in sets), or dividing (splitting up) the silverware or plates. Fractions could be presented as one way of recording partial quantities by cutting up the left over food and demonstrating how each part could be represented. Decimals would follow in a similar line. The table itself presented the perfect springboard for presenting shapes and geometry. Since algebra is just a way of generalizing about quantities, we could really use an a or an x to represent the various objects on the table—or about the height of the table. Economics and statistics, as well as linear graphing and calculus, would come into play if we began to talk about the process of growing the food and selling it to the restaurant…

I finally had to stop myself in amazement. Who would have thought that just a simple dinner table could prove the perfect classroom?

## Online Worksheets/Activities

Wanted to share with you all a math site I found the other day that offers a whole collection of free math worksheets on a variety of different math concepts. The worksheets are pretty much just paper drills, but those of you who are trying to assemble your own curriculum using an assortment of different resources may find this helpful. You could use a problem or two from here to provide the “drill” part of your curriculum, using practical math resources/real-life settings to do the majority of your teaching. http://www.math-drills.com

And since writing the above, I found another site, http://score.kings.k12.ca.us/lessons.html, that offers a variety of mock situation suggestions you can use to help bring math into real-life. Don’t forget to point out to your child that math really is just a way of recording God’s creation, and as such is useful in helping us with the tasks He has given us to do.

## Oh, no! I’m Out of Ideas/Fall Suggestions

Have you ever looked at your child’s math lesson and thought, I have no clue how to teach this. I’m clean out of ideas? If so, you’re not alone. We all face times when we’re simply not sure what to do or how to present something.

A few weeks ago, I knew I needed to write a blog entry on this blog, but I just couldn’t think of anything helpful to say. After a staring at my monitor blankly for a little while, I got up and headed out on a walk. While I walked, I prayed. What was something fall-related parents could do with their children? A leaf just starting to turn colors caught my eye. I wonder if there’s any way to integrate math with the falling leaves? I wondered.

Coming home, I decided to do a little search online on “changing leaves.” I later refined the search to “leaves math.” I found a wide variety of websites offering suggestions for ways to use leaves and math. Although I didn’t like many of the ideas as they were, they inspired other ideas.

For example, http://content.scholastic.com/browse/article.jsp?id=11901 had some interesting facts about leaves, such as that maple trees lose about 600,000 leaves each fall! I thought of how fun it would be to have a child try to count all the leaves that fall in your yard, pointing out that while this is impossible for man it’s not impossible for God. Surely the God who calls each star by name and knows the number of hairs on every person’s head also knows the number of leaves that are on (and falling off) each tree. He is truly MUCH greater than us!!!

In the end, other ideas came, and I didn’t even end up using my leaf idea for my blog. As I hit the post button on my finished blog entry, I thanked the Lord for giving me the ideas that I needed.

If you’re feeling out of ideas, I’d encourage you to take heart. God has the ideas you need. Seek Him. Ask Him. And watch Him give you the inspiration you need in His timing and way.

P.S. If you live in a part of the world where the leaves are changing colors, you may want to do your own search for “leaves math.” Then head out and enjoy the fall weather while you teach your children to use math to explore God’s creation!

## Practical Math Resources

I’ve recently found out about a few practical math resources, and I thought I’d share them with you in case you might find them helpful.

1. http://www.livingmath.net/. This website looked like it had a lot of fun and helpful ideas on integrating math into everyday life.

2. Math on the Level. This new homeschool curriculum, although it doesn’t attempt to present a biblical world view towards math, it does a great job teaching math from real-life settings. It’s format was also very unique and more flexible/easy to modify than others I have seen. I posted my thoughts on https://www.christianperspective.net/math/reviews

3. Arithmetic for Parents. This book by Israeli math teacher Ron Aharoni came recommended to me, and I’ve been enjoying reading it. The book offers a lot of practical ideas about how to teach various concepts as useful tools. It’s available at http://www.sumizdat.org/.

## Organizing

A few weeks ago, I had the opportunity to attend a workshop on organization. Now, slowly but surely, I’ve been organizing my office. The first weekend after the workshop, I tackled all my math notes by organizing them into sets. This organization process reminded me of something I wanted to share here with you all.

As a student, I always used to struggle when my math book began talking about counting numbers, whole numbers, real numbers, irrational numbers, and other number groups. Who determined that these numbers were real numbers, while these other numbers were irrational ones? And what did it matter?

After I began to grasp the biblical worldview in math, I went back and looked at these different number groups and discovered that they, too, were really useful ways of recording God’s creation!

Whole numbers, real numbers, etc., are just fancy names we gives to numbers with certain characteristics. Just like Adam used names to describe the animals, we use these names to describe and sort the quantities God placed around us. Now, we could have used different names. We could have used whole numbers to refer to numbers with different characteristics. These names in themselves are not absolutes; they’re just names that we’ve adopted to help us easily refer to different types of numbers.

Referring to different types of numbers comes in handy. There’s no way any human being could ever remember every single number that can’t be divided by two. But if we refer to all these numbers as odd numbers and learn the characteristics of odd numbers, then we’ll be able to recognize whether or not a number can be divided by two fairly easily. We group numbers into sets because we can’t memorize or keep track of every single number and its properties–only God can do that!

One way you could help your child really see number sets as a way of sorting quantities is to have him organize his closet/bedroom. As you help your child organize his room you could teach math! Tell your child about how, when organizing, we sort objects based on their properties. Have him sort things in his room. For example, we might sort clothes by color (red shirts will go in this drawer, blue shirts in this one, etc.) or by type (long sleeve shirts go in this drawer, short sleeve shirts go in this one, etc.).

Explain to your child that, just like we put like things together to organize, mathematicians have organized numbers based on their properties. Whole numbers, real numbers, etc., all have properties that make them different and unique. We can therefore put them in different “sets”—the mathematical name used to refer to the different “piles” used to organize numbers. Having these “piles” or sets helps us easily refer to numbers with specific attributes, much like the name “long sleeve shirt” helps us easily refer to a certain type of shirt.

There are also a lot of household chores that you can use to teach the concept of sets to your children. Setting the table, folding the wash (by child or by item), or even, putting away the groceries (canned goods, refrigerator goods, freezer goods, etc). Just think–your math class can actually help you get caught up on your housework this week! More importantly, it can help your child see each aspect of math–even number groups/sets–as a way of sorting/recording God’s creation.