As we celebrated our nation’s 250th birthday last week, many went out to see fireworks. Now’s a great time to stop and think about all the math that goes into planning a firework display, along with the wonder of how God governs creation consistently enough that fireworks are even possible (which math helps us see).
Let’s start with planning the display. Here are a few examples of some math involved!
Costs – Say a city has a $100,000 budget for a firework display. That cost might need to include paying a band, bringing in police officers, and other general costs in addition to the costs of the fireworks themselves. Basic math would help compare different options. For example, say they’re selecting different fireworks for a certain 30 seconds of the show.
How much would ten $50 fireworks cost (Answer: 10 x $50 = $500)?
How much more would it be to get 8 of $75 larger fireworks? (Answer: 8 x $75 = $600; $600 – $500 = $100)
Comparing – If one firework show shot off 100,000 fireworks altogether and another did 25,000, how much more did the one do than the other? (Answer: 100,000 – 25,000 = 75,000)
Buying Fireworks on Clearance – If you buy a $50 firework on a 40% off clearance, how much would you spend? (Answer: 40% of $50 is $20, or 0.40 x $50 = $20. Thus the final price is $50 – $20 = $30.)
Math is indeed a useful tool that helps with real-life tasks. Now let’s stop and think about the fireworks themselves, as fireworks not only require lots of math in their development and deployment, butour ability to put on firework displays points to our amazing Creator.
Fireworks contain carefully measured mixtures of chemicals. Different compounds containing metal produce different colors when heated, and manufacturers must mix them in just the right proportions (which we study in math!) to create bright, reliable colors. Manufacturers can only declare with confidence that a firework is red, blue, etc., because the same combinations of chemicals consistently produce the same reactions. And why do they do that? Because God created and sustains a consistent world!
Fireworks also include a fuse—a material that burns while the firework travels up so that the shell bursts safely high in the air and not near the ground. Manufacturers have to observe how quickly different materials they could use as a fuse burn up, and then select the right material and length to make the explosion at the desired point. Designing a fuse involves math, and again relies on materials burning in a consistent fashion.
Example: Suppose a fuse burns at 24 seconds per foot. How many feet of fuse are needed for it to burn for 6 seconds? (Answer:24 sec. was divided by 4 to get 6 sec.; 1 ft. divided by 4 is 0.25 ft. Answer is 0.25 ft., or 3 in.)
How do we know what angle to launch a firework? Or how high it will travel before bursting? Again, we observe the consistencies God created and sustains in creation, such as gravity and the predictable path of a projectile. These consistencies can be described algebraically. Computers use lots of advanced math to calculate when to launch each firework and at what angle so the burst occurs at the desired place and exactly the right moment in the music. This is only possible because of the consistent way God governs creation! If He were not holding this world together in a consistent fashion, we’d have no reason to expect fireworks to go off in a consistent way.
There’s much, much more we could talk about with fireworks, such as the lift charge needed to get the firework off the ground or the arrangement of stars in a shell, but hopefully that gives you a glimpse into how math helps us with fireworks…and how our ability to use math like this relies on God’s faithfulness in holding all things together. As you think back on the Fourth of July with your children, remember to praise the great Creator and Sustainer of all. And know that the math you learn applies outside of a textbook!
Looking for next year’s math curriculum? Check out our middle and high school courses that show students the purpose to math and how it points to the Creator at every turn! And don’t miss our supplemental resources that you can use with younger ones to bring math to life for them.
In helping a friend plan her wedding, I was recently reminded of how math applies in planning an event. Here are a few ideas to share with your students.
A. Total cost given per-person cost: If the caterer charges $29.50 a guest, and you have 80 guests, how much will food cost? If you increase to 90 guests, how much will your catering cost go up?
B. Total with tip: If your total caterer bill is $2,500, and you add a 20% gratuity off the $2,500, how much would the total with tip be?
C. Here is what you’ve spent on center table decorations: $41.23, $32.12, $20.15, and $50.60. If all that decorates 5 tables, what is the cost per table?
D. The baker charges $356 for a cake that feeds 50. What is the cost per person?
Always remember, math is a useful tool that helps us with real-life tasks!
Solutions: A. $29.50 x 80 =$2,360.00; if increase to 90, total is $29.50 x 90=$2,655.00; difference is $2,655.00-$2,360.00=$295.00. B. 1.20 x $2,500 = $3,000 (Multiplied by 1.20 to include the 20% tip, as 100% + 20% = 120% = 1.20; could also have found 20% and added that to $2,500). C. $41.23+$32.12+$20.15+$50.60=$144.10; $144.10/5=$28.82 D. $356/50=$7.12
Need a curriculum that shows students every concept as a useful tool? Check out Kate’s Middle School and High School curriculum, along with elementary supplements that can enrich whatever curriculum you use. Enjoy 30% off most materials through the end of April!
Watching Christmas light shows has become a holiday tradition for many. But few stop to think of all the math that goes into these spectacular displays.
Strands per Outlet or Lights per Circuit
When hanging a lot of lights, you want to make sure you don’t blow a circuit! Bonfe offers this formula to help you figure out how many strands can go in each outlet: # of strands = circuit amperage ÷ strand amperage. We could also use algebra to write this same thing like this: n = c/s or y = z/x. We’re just using letters to stand for the different values, and a fraction to stand for division. Another site says to only use up to 80% of what your circuit can handle as a safety factor. Notice the percents!
Timing
Some light shows include timing with music. Syncing music with a light show requires precision that can be accomplished through various software programs that use math to determine when different effects and lights turn on and off at different moments in the show. In this case, programming the time in seconds that the lights sync with the music includes math!
Cost & Number
The financial cost of Christmas decorations can add up fast! Setting a budget and keeping track of how much money is spent uses addition and subtraction…as well as some rounding skills to keep an approximate total in the store. And how many lights do you need to decorate that garage anyway? Math can help you answer that!
Remember…
As you drive by light displays this year, remember that math applies outside of a textbook. It even aids with Christmas lights.
More importantly, think about why math applies. Why can we even know how many amps a string of light will require? It’s because electricity operates in a consistent fashion. Why? Because God created and sustains an orderly universe.
And that same Creator and Sustainer became a man to make a way for us to know Him by paying for our sin Himself.
Now that’s something to remember this Christmas and always.
Many of us have been fascinated at one time or another by tiny lights dotting the backyard on summer nights. Fireflies, lightning bugs, or whatever you call them are fascinating to watch…and perhaps try to catch. What many don’t stop to think about, however, is how amazingly designed these creatures are.
Light Bulbs
To better understand the marvels of fireflies, let’s pause and take a look at light bulbs for a minute. If you’ve ever put your hand near a light bulb while it’s on, you know that in addition to light, light bulbs produce heat (most of which is infrared radiation). We can use math to measure what’s called the efficacy of the light bulb—that is, how much light they produce compared to how much power they use. The higher the efficacy, the less power is being wasted in heat.
To find the efficacy, divide the lumens produced (a lumen is a unit of measure for light) by the watts (a watt is a unit of measure for power). We could write that as a formula like this:
Or if we wanted to save our hands and use letters, we could write it like this:
We could even use a fraction to represent the division:
Note that we just used algebra! We’re simply using letters as symbols to stand for the efficacy, lumens, and watts of any light bulb. We can write formulas like this because God created and sustains a consistent universe.
If you run to your closet and pull out a package of light bulbs, you can measure the efficacy by inserting the lumens and watts listed on the bulb into the formula. For example, let’s consider a bulb I have in a lamp in my office. It says it is a 53 W (W is an abbreviation for watt) bulb that produces 850 lumens. Leaving units off for simplicity and inserting these numbers into the formula, we’d get this:
Fun assignment: Find the efficacy of the light bulbs listed in the table on step 6 of this page. Notice that some bulbs are more efficient than others! Then take a look at this chart showing the efficacy of different bulbs.
The higher the efficacy, the more the power is going into making light and less into making heat/non-visible radiation. Some bulbs produce more light with less power than others, but they all end up wasting a good deal of power. According to what I could find, an ideal green light that doesn’t waste power in heat would have an efficacy of about 683. That’s a much greater number than the 16.04 efficacy of the light bulb in my office!
What percent of the power is actually used to product light? To find that, we need to find the efficiency, which we’ll define “as the ratio of the total light power to the total power.”[i] The efficiency can be rather tricky to figure out, as light bulbs don’t come labeled with how much power goes towards light, but one way to approximate it is to divide the efficacy (in lumens per watt) of the bulb by 683 (the efficacy of ideal green light, which is the highest efficacy possible and thus can be used to approximate the total power while we use the efficacy to approximate the power used to produce the light).
Plugging in the efficacy we found for my office light bulb would give us the following:
If we express this as a percent by multiplying by 100 and adding a % sign, we’d get approximately 2.35%. That means only about 2.35% of the power this bulb uses goes towards making a light; in other words, the other 97.65% goes toward making heat/non-visible radiation. We would say this light bulb is about 2.35% efficient.
Fun assignment: Check out this table to view the efficiency of various lights…and calculate the efficiency of some lightbulbs in your closet by first finding the efficacy, then dividing that by 683, and finally expressing that result as a percent. Notice all the math you’re using!
Fireflies
Now, fireflies don’t plug into an electrical outlet or battery to produce light. Instead, a firefly produces light using chemicals. Yes, that’s right: God created these little bugs to have a mini chemical factory on board! These chemicals prove useful to man as well—one of the chemicals used has been useful in “biomedical research.”[ii]
The chemical reaction in the firefly produces energy…which is what powers their lights. When we talk about the efficiency of the lights fireflies produce, we’ll talk in terms of energy. Note, though, that power and energy are related–power is the rate of use of energy. We could say that efficient bulbs use less power and less energy.
Unlike man-made light bulbs, the firefly produces light without wasting much energy at all—sources indicate almost 100% of the energy turns into light[iii]—in other words, it’s almost 100% efficient. Compared to the 2.35% efficiency we found for the light bulb in my office, that’s amazing! Yet if the firefly didn’t produce light efficiently, they’d heat up and burn to death.[iv] Instead, God designed them to produce “the most efficient light in the world.”[v]
Conclusion
Next time you head outside and see the light from fireflies, ponder God’s magnificent design. Think about the wisdom and care God took with His creation, even giving us light displays at night made from tiny living chemical factories. And remember that math helps us explore, describe, and better appreciate His amazing world…as well as to improve inventions (like light bulbs). We’re able to explore and design because God made us in His image. That same God wants each of us to know Him and live in His presence forever.
Math probably isn’t the first thing you’re thinking about when you don’t feel well. Yet you might be surprised at how much you may find yourself using math when facing an illness.
When Can I Take ____ Next? When can I take that next does of pain medicine/cough suppressant/allergy pill? That question is on our minds a lot when battling a disease. And we use math to figure it out! If we took medicine at 10 am and we can take a pill every 4 hours, then we next need to take it at 2 pm, as that is 4 hours later. Keeping track of time’s passing is an application of math—an application, by the way, that is based on the earth’s rotation that God set up. Our 24-hour clock ties with the pattern of night and day He created.
How Many Days Left? How many days till I feel better if the cold started 3 days ago and typically lasts 7 days? 7 – 3 = 4, so hopefully just 4. Till what day do I have to quarantine if I was told to quarantine for 10 days starting on the 11th? 10 + 11 = 21, so till the 21st.
How Much Should I Take? If I was told to drink 70% of my body weight in ounces per day and I weigh 120 pounds, how many ounces of water do I need to drink? 0.7 x 120 = 84, so 84 ounces each day. If the glasses I use hold about 10 ounces of water, how many glasses does that mean? Well 84 10 = 8.4, so 8 and almost a half of those glasses. Or say I was told to take 4,000 IUs (international units) of Vitamin D a day, and I found a liquid that contains 500 IUs in a drop. How many drops would I need to take? Well, 4,000 500 is 8, so 8 drops.
Note that we perform simple math all the time without thinking about it. And aren’t you glad that to figure out how many glasses of water equals 84 ounces you don’t have to actually measure out 84 ounces and pour it into 8 and a half glasses? Instead, you can use math to know that 84 divided 10 equals 8.4. You can only do that, though, because creation is consistent. Division operates consistently. Why? Because a faithful, consistent God is holding creation together. Addition, subtraction, multiplication, division, and the other “fixed order” He created continue because He is a covenant-keeping God (Jeremiah 33:25-26 ESV). Now that’s a good thing to remember next time you’re sneezing your way through a cold.
Why did God create snot? I was recently asked that question by a little girl, and it’s a great one! Sickness wasn’t part of God’s original creation. There was no sickness or death until man sinned, rebelling against our Creator. Now we live in a fallen world. But praise God He designed our bodies to produce snot to help rid our bodies of bacteria and viruses. And praise Him that He sent Jesus to buy us back for Himself so that one day, if our faith is in Jesus, we’ll be in a world with no more sickness again.
With the Major League Baseball World Series well under way, I thought it would be a good time to take a look at the math involved in baseball. We’ll also see how all of it reminds us to praise the Creator.
Basic Numbers
Some of math’s applications to baseball are both obvious and basic. How many pitches a pitcher has pitched (called the “pitch count”), the strike and ball counts, the scores (called number of “runs”), the number of innings, the number of players, the numbers on the players’ jerseys—all these and more are examples of basic counting and numbers in action. How many innings are left after the 3rd finishes (assuming the game is not tied at the end)? That can be found using subtraction; taking 9 (the total innings with no tie game) – 3 (the innings completed) leaves the answer of 6 left. Do that same math after the top of the 3rd instead (meaning only half an inning has completed), and you get 9 – 2 ½ = 6 ½. Notice the fractional amount.
Basic Geometry
Baseball also has lots of examples of basic geometry in action. The shapes of the bases (squares), home plate (an irregular pentagon), ball (sphere), and infield (square or diamond) are all basic shapes familiar to many students. The perimeter of the infield would let us know how far a runner runs when he hits a home run; the volume of the ball could be useful if we needed to find the density of the ball (which would require basic algebra—the formula is listed below, where m is the mass and V the volume.
Statistics
When watching a baseball game, lots of statistics appear on screen. These statistics are calculated using math!
For example, batting averages are the result of taking the total hits divided by the total at bats, or total hits ÷ total at bats. (Note that total at bats “does not include walks, hits by pitch, or sacrifices” https://www.wikihow.com/Calculate-a-Batting-Average.) Notice this requires both counting up the total hits and total at bats, and then uses division!
Here’s a site that explains the math behind various other statistics used: https://www.mlb.com/glossary/standard-stats/earned-run-average. If you’re teaching elementary math, you might want to check it out and make up some problems for your child to use. For example, knowing that batting averages are found using the formula total hits ÷ total at bats, make up some numbers for total hits and total at bats. (Note that the total hits will typically be less than at bats…unless a player hits the ball every time!) Say a player hit the ball 5 times and had 25 at bats; a student can calculate then that his batting average is 5 divided by 25, or 0.2, by plugging 5 in for total hits and 25 for total at bats.
You might want to point out that whenever statistics are used, it’s important to ask questions about them, as they can easily be presented in a misleading way. While this isn’t really an issue in baseball (unless someone misunderstands the stat), baseball can be used to illustrate how statistics can be twisted to help students learn to discern through the bombardment of statistics we encounter in the news, advertisements, etc., where they ae often twisted.
For example, say it’s a player’s first time at bat ever in the major leagues. He hits the ball during that at bat. His batting average the next time he steps up to the plate is 1 (he’s had 1 hit and 1 at bat), which equals 1. If you just see a batting average of 1 you might think he’s an amazing player…but we don’t really know that yet. This illustrates a general rule in statistics that the more data you have to calculate an average off, a better reflection of reality is the average is. We need a lot more at bats than 1 to really know how good a player is based on their batting average. In a similar vein, batting averages can be calculated over a player’s lifetime, for a series, or for a game. The point is that it’s always important to understand how a stat is calculated and what it’s really saying.
Algebra and Calculus
We can use algebra in calculating perimeter, area, and volume (the formulas are often written algebraically). We can also use algebra to figure out different facts from various stats. For example, if we know the formula for a batting average and one of the other values in the formula, we can use algebra to find the other value. Say we know a batting average over 100 at bats is 0.2. That means this:
Since we typically use letters to stand for unknowns in algebra, let’s replace total hits with an x. We’ll also replace the with a fraction line, as that’s the typical way to show division in algebra.
Now, because of the faithful, consistent way God governs creation, we can multiply both sides by 100 to solve for , thereby seeing that this player must have had a total of 20 hits.
Algebra (and calculus too) can also help us explore the physics behind baseball. For example, the speed of a pitch is cited throughout a game. While that speed is measured using a radar gun (a device built by using math to both understand and then design around the consistent way God governs creation), you could explain to your student how we could use algebra to calculate the approximate time the ball was in the air given that speed.
We know from physics that distance = speed • time, or d = st. If we solve this equation for time (t) by dividing both sides by s, we get d/s = t. According to the MLB, the distance from the pitcher’s mound and home base is 60 ft 6 in (approximately 0.011458 mi). Let’s say a pitcher pitches a ball that goes 90 miles per hour (mi/hr). Approximately how long was the ball in the air before it crossed home plate? Let’s plug in 0.011458 mi for distance (d) and 90 for the speed (s).
Formula solved for time (t):
Plugging in the values and simplifying:
Converting that to seconds using the knowledge that there are 60 seconds in a minute and 60 minutes in an hour, we’d get approximately 0.45832 seconds for the time the ball was in the air, so not quite half a second. That’s not long!
While I won’t go into it in this post, there’s much more that we could use algebra and calculus to explore in baseball, including exploring the force the batter has to apply to the bat in different situations. Check out Chegg’s “Baseball and Calculus” if you’d like to explore calculus (plus really a lot of algebra too) and baseball further.
Pausing to Praise
Think about the fact that in less than half a second (see the Algebra and Calculus section above), a batter is able to process a pitch and decide if and how to swing.
When I was going through a concussion recovery, I remember reading an article in the Answers magazine about how our brain is designed to learn. God has wired us so that if we practice a skill over and over again, our brain will actually change in response, allowing batters to train their brains to respond to pitches. Pause for just one minute to ponder that—and praise the God that made your incredible brain.
“I praise you, for I am fearfully and wonderfully made. Wonderful are your works; my soul knows it very well.” Psalm 139:14 (ESV)
Also think about all the math that is used in simply setting up a baseball field where the bases are the right distance from each other…and in analyzing a game (let alone a season). We tend to treat statistics as commonplace, but they’re really shouting out at us that we live in a consistent universe held together by a consistent God. After all, they’re found by using division and other math operations…which hold true because day after day, century after century, millennium after millennium, God holds creation together consistently.
Indeed, we need to take a moment to pause to praise—and to reflect on the fact that knowing our amazing Creator is the greatest goal we could ever run after. We achieve salvation from sin and a relationship with God by receiving “the righteousness from God that depends on faith” (Philippians 3:9 ESV), but the Christian life doesn’t end there. Like athletes chase after goals, we’re called to chase after living more and more like we will be one day in Heaven—Christ’s completely.
“But one thing I do: forgetting what lies behind and straining forward to what lies ahead, I press on toward the goal for the prize of the upward call of God in Christ Jesus.” Philippians 3:13b-14 (ESV)
Baseball is one of countless examples of math in action students explore in the Principles series. If you’re looking for a curriculum that both shows math’s usefulness and points math to the Creator, be sure to check out the biblical math materials we’ve put together.
My husband and I recently moved into our first home together…and we’ve done a lot of math along the way! Below are a few ways math is used while moving into a new home.
Buying the Home – Buying a home requires a lot of paperwork…some of which lists numbers such as your down payment, closing fees you owe, closing fees the seller owes, etc. All those numbers need checked (my husband actually found an error in one draft they had to fix), which requires basic adding and subtracting on a calculator with decimal numbers (and knowing what to add/subtract).
Unloading Furniture – Where should we put ______? Determining where furniture goes isn’t always as easy as it sounds. But rather than lifting it and trying it in lots of places (which can be really hard on heavy pieces), measuring the piece and measuring the room (or part of the room) can help figure out where to put it without breaking your back. Addition and subtraction (including of fractions), as well as unit conversion, also come into play. For example, if the window is 45 1/2 inches high and the table comes up 30 3/16 inches, how much space will be between the table and the window? Is it enough for a project that needs to be 5 feet off the floor to sit on?
Installing Curtains…and Hanging Pictures – Buying a new home, we had a lot of things we needed to install, from curtains to towel racks to cabinet doorknobs. And then there’s hanging pictures. Measuring helps you know where to drill the hole for the screw/put the nail so it all ends up at the right height. Not only does measuring and basic arithmetic help out here, but so do problem-solving skills. When my husband installed the kitchen drawer handles, he had to think through how to get them centered. After all, neither screw needed to be in the center of the cabinet—the center of the handle needed to be there, but the screws were on the sides of the handle. He had to think through that if he measured the length of the handle and divided it in 2, he’d find the distance from the center of the cabinet on either side he should put the screws in order to have the handle centered.
Buying Furniture – What size rug do we need to buy? Will that loveseat fit? If we put the loveseat there, will the recliner still have room to recline back? All of this requires measuring, addition, and subtraction (including sometimes of fractions). When will my furniture arrive if shipping takes 2 weeks? That again requires addition!
Budgeting – How much are we spending? How much do we have left to spend? Again, addition and subtraction (with decimal numbers this time) in action!
Math is far from a mere textbook exercise—it helps us complete tasks. Those are only a few tasks math helped my husband and I complete lately.
As you teach math, be sure to let your child use math outside of a textbook too. Doing a home improvement project? Let them help you figure out the cost, make measurements, etc. (If you don’t have a project, you could make a hypothetical one to work on—that could be as simple as seeing if a certain piece of furniture would fit in another room, or how many gallons of paint would be needed to repaint a room (and how much it would cost at $24.99 a gallon).) Remind them that God gave us work to do here on earth, and math is a tool to help us in that work. And it’s a tool we can use while praising the Creator and working to His glory.
Looking for Good Math Materials? Revealing Arithmetic helps you transform the younger grades (plus we sell a complete elementary program), Principles of Mathematics is a complete program (with an optional ecourse) for junior high, we just released an ecourse to go alongside Jacob’s algebra, and we’re getting into the final stages of Algebra 2 (watch for it in early 2021, Lord willing).
The Bible contains a lot of measurements. For example, in Revelation, we’re told that the New Jerusalem’s “length the same as its width. And he measured the city with his rod, 12,000 stadia. Its length and width and height are equal.” (Rev. 21:16 ESV). The questions below can be used to help students explore these measurements and get a deeper appreciation for just how incredibly big the city God is preparing for His people really is.
A stadia is about 600 feet (this is a conservative estimate–others say slightly higher). Knowing this, how many feet is 12,000 stadia?[1]
To put that in perspective, convert the number of feet you found to miles, given that 1 mile is 5,280 feet.
Now, to get a better sense of that distance, compare it to the distance from Boston, MA, to Orlando, FL, which Google maps estimates the shortest driving distance (at the time I checked it) to be 1,287 miles. Which is greater, and by how many?
Students should find that the measurement of the New Jerusalem is actually greater than that from Boston to Orlando! It’s hard for us to even picture a city of that size–and that’s the dimension in each direction (we’re told its length, width, and height are equal–that is, it’s a cube!). Henry Morris proposes that we, like angels, will no longer be constrained by gravity but rather be able to move around vertically as well,[2] making the height measurement extra relevant. The point is that this is an amazingly large city, and math skills help us get a better understanding of its magnitude.
After completing the exercise with students, read Revelation 21 together. Talk about how the New Jerusalem will be perfect–no suffering or death. But how to get into it requires being born again. Revelation 21:9 (ESV) makes it clear that all cowardly, faithless, detestable, murderers, sexually immoral, sorcerers, idolaters, and all liars will be in “the lake that burns with fire and sulfur.” If we’re honest, we’re all those apart from Christ. Jesus made it clear God counts the thoughts of our hearts as well as our actions; in His eyes, hatred makes us guilty of murder, lust of sexual immorality (Matthew 5:21-30). And we’ve all told a lie–and been an idolater, looking to something other than God for what we should have looked to Him for. While discussing how sin keeps us from God’s Kingdom, 1 Corinthians reminds us that “And such were some of you. But you were washed, you were sanctified, you were justified in the name of the Lord Jesus Christ and by the Spirit of our God” (1 Corinthians 6:11 ESV).
The only way we can enter God’s coming perfect kingdom is through Jesus, being born again in Him (John 3:7). We need His righteousness to enter His perfect Kingdom. Challenge students to make sure they’ve put their trust in Jesus…and then to tell someone about what He has done to save us from eternal punishment and give us eternal life in a perfect world with Him forever.[3]
Algebra 2 Update
The publisher is working hard on formatting the Algebra 2 program, and we’ve almost finished the optional eCourse to go with the lessons. The process was delayed several months due in part to COVID, but is now in full gear. We don’t have an exact publication date yet, but we hope it will be completed sometime later this year. We apologize for the delay, appreciate your understanding, and would cherish your prayers.
[1] Some versions translate this a furlong, which would be 660 feet. We used the more conservative distance. [2] Henry M. Morris, The Revelation Record (Tyndale, 1983), p. 451. [3] See LivingWaters.com for evangelism materials and ideas.
Meet the Author
I’m a sinner saved by grace through Jesus Christ. I’m also a writer—I’ve been writing ever since I could hold a pen. I… Meet Katherine [Loop] Hannon.
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