The featured Math Game this week involves no arithmetic, no counting, and no calculation. But it is an excellent (and enjoyable) introduction to the concept of plotting points on the Cartesian plane. Granted, the orientation of the vertical axis in the game is inverted from the usual Cartesian y-axis: the index increases, rather than decreases, as one moves downward. And points are plotted in the open squares, rather than on the intersections of the grid lines. But these are small issues, and we've found they cause no problems for children as they transition to the real Cartesian plane.
VariationsNow, if you really want to build familiarity with the actual Cartesian plane, one variation is to use a paper printout of the xy plane. (They're available on TeachersPayTeachers.com for $2.00 or less.) Or, you can make your own xy plane, with coordinate limits of your choice. You can make it as large as you want. And I would recommend you go ahead and center your printout at the origin, so that both positive and negative coordinates are used. It's never too early to introduce the concept of negative numbers.When you're ready to play, simply draw an oval of the appropriate size around the string of points where you want your ship to be. When your opponent scores a "hit", place an X on the point within the oval where the "hit" occurred. To begin with, your children will fire in a somewhat random fashion, naming points as the mood strikes them. But as they mature, you can help them develop a more systematic approach. Teach them to cover the board with a coarse "net" of shots. Then, if the small ships still escape undetected, gradually refine the "net" by subdividing the open sections. When your children are somewhat more sophisticated, start getting creative with the game, and customize your own weapons. I've played the game sometimes where each player has several mines available, of varying size. A mine has an area effect. When using it, you specify the mine's location, and any portion of a ship within the detonation radius of the mine is blasted. Of course, your opponent has to notify you of the mine's effect, and where the "hits" occurred. I've also allowed a certain number of sonar "pings" per player. When a using a sonar "ping", you specify a square region on the board, and your opponent has to notify you of the presence of any ships within such a square. They don't have to give you the coordinates of the ships. They just have to say "Yes, I have a ship somewhere in there", or "Nope, keep searching". And if they have multiple ships in the square, they have to tell you how many ships there are. In summary, Battleship is a wonderful game. Lots of fun, lots of variations. And it's great for teaching the plotting of points in the plane.
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Parents, suppose your child approaches you and asks for an allowance, subject to the following conditions:
- The child will do all dishwashing, vacuuming, and outside yard work for one year. Yard work includes mowing, weed-eating, and gardening. You won't have to do any of it.
- The child will have Sundays off from vacuuming, and yardwork, but will still do dishes.
- The child will only be paid for the first 8 weeks, 6 days a week, Monday through Saturday. The rest of the year's labor is completely free.
- The child's payment for the first day will be one penny. (Yes, just one cent.)
- The second day's payment will be doubled to two pennies.
- The third day's payment will be doubled again to four pennies.
- Payment will continue doubling at this rate for the remainder of the 8 weeks (Sundays excluded).
Absolutely NOT!!!Should you accept the Proposal?Answer: Some reasonably intelligent folks have answered this question by saying, "Well, maybe if you were a millionaire, and had plenty to spare." Nope. Those folks are underestimating the power of exponents. It wouldn't matter if you were the richest person on earth. In fact, it wouldn't matter if you were as wealthy as the top 50 richest people in the world..... combined. Why? Because your total bill at the end of the 8 weeks would be more than $2.8 !!!! To be precise, the actual bill would be $2,814,749,767,106.55. (Mind you, don't forget the 55 cents - wouldn't want to cheat the poor kid, would you?)trillionPretty hefty allowance, eh? That's the power of exponents. After the first week, you'd be feeling good, thinking you'd made a sweetheart deal. Actually, you might even feel a bit guilty about taking advantage of your kid. The first week's total payment would be the grand sum of 63 cents. But then, at the end of the second week, some question marks might be popping up in your head. After all, the second week's payment would be $40.32. Seems a bit steep for an allowance. And the worst part is, the first day of the third week will earn a paycheck of $40.96. That's just the first See, the paycheck for each day is actually one penny more than all the previous paychecks day! .combinedNow do you understand the power of exponents?Okay, so if you understand exponents, how should you counter your kid's payment proposal? I mean, you'd love the extra help around the house, and you'd like to instill appreciation in your child for a well-earned paycheck. But I think you'd agree that $3 trillion is a bit excessive. Well, one possibility is to answer this way: "Johnny (or Jane), I really appreciate your offer, but I'd hate to work you too hard at your age. You know, 'all work and no play' and all that. So let's keep all of your conditions, except one. I want you to have five Sundays a week, and the vacuuming and outside work (and the paychecks) will only be two days a week." Will that be enough to save your financial solvency? Well, the total bill now comes to $655.35. That's a fair amount of money. But remember, the dishes will be done every day for a year. Vacuuming twice a week. Yard work twice a week (and if you have a big yard, think how much a landscaping service would charge). And don't forget the gardening! Still think it's too steep? Then grow a larger garden, and the kid can pay his own salary with the surplus vegetables. Now, if you really want to skin your kid (as punishment for trying to skin you), you can take it one step further. Allow vacuuming and yard work (and paychecks) on only one day a week. That should be enough to get the weekly mowing and weedeating done. Also, you'll get the vacuuming for that day, and maybe a little gardening. Plus, don't forget you'll have all the dishes washed day of the year. every And how much would that set you back? Ha! A grand total of two dollars and fifty-five cents..... for the entire Now that's a sweetheart deal!!year!!!Ah, the power of exponents.
Back in the good old days, when I was teaching Calculus to undergrads, I remember two types of students: those who could calculate, and those who couldn't. Now, certainly there are other skills which are important for the study of higher mathematics. There's creativity, and imagination, both of which feed into visualization and abstraction. But without the ability to calculate, to do simple arithmetic quickly and easily, the other skills are handicapped.
I had students who could grasp difficult concepts, could creatively plan methods of attack on complex problems, or could abstract fundamental truths from seemingly unrelated data. But they couldn't bring those tools to bear in their full strength, because they had a weak foundation in arithmetic. And they were often slow in following my reasoning on the board, because they were still hung up on a simple factoring that occurred two paragraphs earlier ("Dr. Fulton, where did that come from?"). I had students who got Calculus. They truly understood the conceptual picture. But they struggled solving problems, simply because of arithmetic. Why? I don't know exactly. I can only guess that they were allowed to use calculators throughout grade school, and they missed out on the reflexive training for arithmetic. That's a heartbreaking situation, because it's not easily fixable at a later stage. No Calculus student wants to be told to go back and review multiplication tables.
We mentioned one flash card technique that we've used in our "Shoot-the-Number" post, involving Nerf guns. It involved the child saying the answer, and then shooting it on the number chart if he was right. As he improves, make him get 5 in a row before he can take a shot. Then make him get 5 in a row in under 10 seconds. It's like weightlifting. Keep making it harder, tougher, faster. You want the arithmetic to be reflexive.
Another technique we've used is the trampoline. A child gets to keep jumping on the trampoline, as long as they're doing the flash cards and getting them right. Once they've missed, say, three cards, it's time to get off. Choose an activity your kid enjoys, and do likewise. If you have children of different age levels, you can teach multiple kids at once. We just did this yesterday. One child held up a card, a younger child pronounced the numbers on the card (for number recognition), and an older child answered the multiplication problem. Meanwhile, the child holding up the card was also getting a review. Sometimes, it is useful to call out the card to your child, without letting them see it. It's different, and it will slow them down at first. But it's good. It's important to be able to do mathematics without necessarily relying upon optical signals. It builds visualization. There are many, many variations that can be done with flash cards. Be creative. But be consistent. Use them regularly. They're a good, cheap tool in building an arithmetical foundation that will last... ...to Calculus, and beyond!
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This puzzle was sent to us by a family member. It requires either a little algebra, or else a little "guess-and-check" work.
We hear it all the time.
"My child's so smart! My child's a genius! He's brilliant! So creative!" And then follows the inevitable... "But I can't teach him anything, because he just won't sit still!" (We use "he" and "him" throughout this post, because this problem overwhelmingly happens to boys - including our own. Our girls were model students for us. They listened attentively to every word. They sat still, and did their work most of the time. We thought we were master teachers. And then our boys came along...)
Now, what should the questions be? Well, it depends on the age of your child. The sky's the limit. Any question with an answer from 1 to 100 will suffice.
Examples:1) When your child is just learning numbers, point to a number on the chart, and have him pronounce it. If he's correct, have him shoot it.2) Flash cards are great for this! Just have your kid tell you the correct answer out loud before shooting, because Nerf guns are rather imprecise.3) Do Flash cards without the cards. Meaning, don't show the card to your child, but instead say it aloud to him. It's a good skill to be able to add or subtract or multiply without seeing the numbers on paper.4) Do word problems. Your child is more likely to fight through the frustration that comes from word problems, if there's the chance to take a shot afterwards.5) This isn't math, but it's a bonus. Use it for geography. Put a world map on the wall (or a country map, or a state map, etc.), and have them shoot certain countries or oceans as you call them out. This is surprisingly fun. So much so, that the teacher has even been known to take up arms a time or two. (Those European countries are downright devilish to hit on a world map. Good luck shooting Switzerland with a Nerf gun.) Okay, now some will say "Yeah, those sound fine, but it's much more efficient to just have them say the answers out loud to me, or write them on paper. We can go much quicker that way." Guess what? They're right! And if your kid is able to sit still and concentrate well enough for your liking, then by all means do what's most efficient. Our girls did great in that regimen. And we intend to get our boys to that point, as well. But at this stage for our boys, well, it just ain't happenin'! And we're gonna do what works. Even if it's not the most efficient technique. Oh, and even if your child doesn't need "Shoot-the-Number" to stay on task, it's kinda fun to offer it anyway on special occasions. Our daughters don't need it, but they're not averse to pulling the trigger, by any means! Just be warned. A stray projectile sometimes (often?) manages to strike the teacher. By accident, of course. If you like our puzzles and explanations, please visit our store and check out our problem-solving and logic puzzle books!This post may contain affiliate links for which we will earn compensation should you choose to make a purchase. We are disclosing this information in accordance with the Federal Trade Commission’s 16 CFR Part 255, Guides Concerning the Use of Endorsements and Testimonials in Advertising. Thank you for your support of The Math Profs.
Four people (Bob, Jim, Ken, and Travis) are participating in a parade. They are each riding a different vehicle (on a float, in a golf cart, on a bike, or in a truck bed), and wearing a different costume (sheep, baseball uniform, clown, or bear). Match each person with the appropriate vehicle and costume. - The four people could be referred to as the person wearing a clown costume, the person wearing the baseball uniform, the person riding in a truck bed, and Jim.
- Another way of referring to the four people is the following: Ken, the person wearing the clown costume, the person wearing the sheep costume, and the person riding the golf cart.
- The person dressed as a baseball player is riding on a float.
- When asked where he would be in the parade, Bob said he wasn’t sure, but he would either be on a float or riding a bike.
Full tutorial found here.
This puzzle was taken directly from Book 3 of our "Grids For Kids" logic puzzle series. If you like our puzzles and explanations, please visit our store and check out our problem-solving and logic puzzle books! Benefits**Number Ordering****General Numeracy**
If your kids like math, it's pretty easy to teach it to them. Self-motivation makes everything run smoothly.
But what if your kids don't like math? Or what if they're so young, they don't even know what math is? How do you build a foundation and instill concepts? Do you just sit 'em down at a table, hand 'em a hundred 4-digit multiplication problems, and tell 'em they're gonna do 'em till they like 'em? Yes, we've tried it. No, it doesn't work. Instead, we've found the best way to teach math is... to teach it without teaching it.
Confused?
Well, let me say it like this. If you can convey certain concepts without having to resort to chair, desk, and chalkboard, then so much the better! Learning is learning, either way. And we've found there's a greater chance of mutual enjoyment (and retention!) if the teaching occurs unconsciously. This is where games come in. Board games, card games, dice games, you name it. Games are fun, and games can teach. (I'm excluding video games because...well...I just am.) Now, whenever I mention "games" and "teaching kids math", someone always says, "Oh, you're talking about Chess, aren't you?" No, I'm not talking about Chess. I don't know why everyone always associates Math with Chess. (Is it because nerds like math, and nerds like Chess, so therefore Math and Chess must be...the same?) Chess is a wonderful game, don't get me wrong. It can help build concentration, encourage strategic thinking, develop visualization, logic, etc. But Chess is not what I would use to build numeracy.
As a child grows older, subtraction comes into play subconsciously, because that helps determine interval widths for placement. I, myself, use a little probability, together with memory of what cards have already been played.
But without a doubt, the primary educational value of this game is ordering. After several days of playing RACK-O, ordering numbers is second nature. And did I mention that it's fun? It is! Actually, I'm quite addicted. I could play it every night. Trust me, it's a lot more fun than teaching "number ordering" on a chalkboard. So take our advice, and use games to supplement your math instruction. Your kid will enjoy it, and you will as well. Do you have a favorite math game? If so, leave a comment telling us the name of the game and why you like it. We are always on the lookout for new favorites!!
If you like our puzzles and explanations, please visit our store and check out our problem-solving and logic puzzle books!
This post may contain affiliate links for which we will earn compensation should you choose to make a purchase. We are disclosing this information in accordance with the Federal Trade Commission’s 16 CFR Part 255, Guides Concerning the Use of Endorsements and Testimonials in Advertising. Thank you for your support of The Math Profs.
1. How many unique license plates are possible?2. Suppose no license plate is allowed to use the letter "O". How many license plates are possible?3. Suppose no repetitions are allowed--which means that no letter or number can be used more than once. How many license plates are possible?4. Suppose no repetitions are allowed, and the letter "M" has to be used. How many license plates are possible?5. Suppose repetitions ARE allowed, but the sequence 211 is NOT allowed. How many license plates are possible?You will find the solution here. To Ponder On Your Own: Would the answer to number 5 change if the disallowed sequence was 111 instead of 211?If you like our puzzles and explanations, please visit our store and check out our problem-solving and logic puzzle books!
A year from now, David's age will be twice Jane's age. Two years ago, David's age was three times Jane's age. How old are David and Jane right now? You will find the solution here. Now, how does that help us? Well, it doesn't….yet. But if we arrange them a little differently into columns, and add each column first, we have \[ \array{ &1 & +& 2&+& \ldots &+ & 99 & + & 100& \\ +&100&+&99&+& \ldots &+& 2 &+&1&\\ \hline \\ =&101&+&101&+& \ldots &+&101&+&101& } \] Do you see what we did? We just rewrote the numbers so that all of the columns would be equal, and then summed each column. Since we know there are \(100 \) columns, we can now use multiplication (much faster than addition) to see that the above is equal to \(100 \times 101=10,100 \). But we're not done yet. In order to make our columns equal, we had to add our sequence twice. So now, to get our final answer, we need to divide \(10,100\) by \(2\) to get \(5050\). [And just remember, a 6-year-old figured this out! Of course, that 6-year-old was none other than Carl Friedrich Gauss, one of the greatest mathematicians of all time. And we really don't know that he was 6, we just know he was in primary school. But still pretty impressive, wouldn't you say?] Now, if you look closely you'll see that there's nothing special about the number 100 in this trick. Any number will work. And you don't even have to remember the steps we took. Let's just make a formula, and then you'll have everything you need. General FormulaSuppose \(N \) is a positive integer, and we want to add the numbers \(1 \) through \(N \), written as \(1+2+ \ldots +(N-1)+N \) As before, let's add them twice and line them up in columns: \[ \array{ &1 & +& 2&+& \ldots &+ & (N-1) & + & N& \\ +&N&+&(N-1)&+& \ldots &+& 2 &+&1&\\ \hline \\ =&(N+1)&+&(N+1)&+& \ldots &+&(N+1)&+&(N+1)& } \] This is equal to \((N+1) \times N \), and since we added everything twice to do our trick, we need to divide by 2. And this gives our formula: \[ 1+2+ \ldots +(N-1)+N= \frac{N \times (N+1)}{2} \]. Go ahead and try it out!! It works every time. And here is some homework for you to try. First, try this one: 1. Calculate \(1+2+ \ldots +37+38 \). Now let's see if you can adapt your thinking to these: 2. Calculate \(2+4+ \ldots +98+100 \). 3. Calculate \(20+21+22+ \ldots +79+80 \). 4. Calculate \( 37+40+43+46+ \ldots +94+97+100\). If you have trouble adapting the formula for those last three, you can always add them twice and line up the columns, right? |
## AuthorsBrian and Melanie Fulton both earned doctoral degrees in mathematics at Virginia Tech. They formerly taught math at the university level, and now run a hobby farm while accuracy-checking collegiate mathematics texts. They homeschool their four children, frequently employing the aid of chicken, dairy goat, cat, and dog tutors. ## Archives## Categories |