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 "ShoottheNumber" 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 "guessandcheck" 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 "ShoottheNumber" 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 problemsolving 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.
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 problemsolving and logic puzzle books!
Benefits
If your kids like math, it's pretty easy to teach it to them. Selfmotivation 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 4digit 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 RACKO, 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 problemsolving 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 allowedwhich 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 problemsolving 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. If you like our puzzles and explanations, please visit our store and check out our problemsolving and logic puzzle books!
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 6yearold figured this out! Of course, that 6yearold 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 Formula Suppose \(N \) is a positive integer, and we want to add the numbers \(1 \) through \(N \), written as \(1+2+ \ldots +(N1)+N \) As before, let's add them twice and line them up in columns: \[ \array{ &1 & +& 2&+& \ldots &+ & (N1) & + & N& \\ +&N&+&(N1)&+& \ldots &+& 2 &+&1&\\ \hline \\ =&(N+1)&+&(N+1)&+& \ldots &+&(N+1)&+&(N+1)& } \] 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: \[ 1+2+ \ldots +(N1)+N= \frac{N \times (N+1)}{2} \]. 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? If you like our puzzles and explanations, please visit our store and check out our problemsolving and logic puzzle books!
Question 1: The correct answer is 4 crayons. If Jeremy picks 2 crayons, they might be the same color. Might. But, then again, they might not. And Jeremy wants a guarantee, so 2 crayons won't cut it. What about 3? Nope. He might end up with only one of each color. Will 4 crayons do it? Why, yes! It will! Even in the worst case scenario, where the first 3 crayons are each a different color, the 4th crayon would have to match one of the first 3 crayons since there are only 3 distinct colors. Question 2: The correct answer is 21 crayons. If Jeremy picks 20 crayons, then it's technically possible for him to have 13 blues, 6 browns, and 1 red. So 20 crayons is still not enough to guarantee at least 2 reds. He would need to grab 21 crayons for the guarantee. Question 3: The correct answer is 35 crayons. If Jeremy picks up 34 crayons, he might have all reds and blues. He would need one more to guarantee he has at least 1 crayon of each color. Question 4: The correct answer is 26 crayons. The first 19 crayons Jeremy grabs could be all blues and browns. In this (unlucky) case, he would have 6 browns and would need to grab 7 more crayons from the box, which now contains only the red crayons. And if you still want more puzzles of a similar type, check out our "How Many Socks?" book, for sale at our store www.themathprofs.com. We'll walk you through the process of solving these types of questions  from questions just like these, slowly building to more general scenarios using variables for the numbers of colors and items.
If you like our puzzles and explanations, please visit our store and check out our problemsolving and logic puzzle books!

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 accuracychecking collegiate mathematics texts. They homeschool their four children, frequently employing the aid of chicken, dairy goat, cat, and dog tutors. ArchivesCategories 